Fatt Bp te keiths eee : eee pees AR BT ER a Fire Ria Pang ms =e ea ES Ne LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN O10 L61 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN L161—O-1096 Digitized by the Internet Archive in 2022 with funding from _- University of Illinois Urbana-Champaign https://archive.org/details/libraryofusefulk01unse ait LIBRARY USEFUL KNOWLEDGE. MATHEMATICS. I. STUDY AND DIFFICULTIES OF EXAMPLES OF THE PROCESSES MATHEMATICS. OF ARITHMETIC AND ARITHMETIC AND ALGEBRA. ALGEBRA, LONDON: BALDWIN AND CRADOCK, PATERNOSTER-ROW. MDCCCXXXVI. LONDON: Printed by WiiL1am Crowes and Sons, Stamford Street, S\0 L6\ vA ‘se 5 ae } CONTENTS. Page Stupy AND DiFFICULTIES OF MATHEMATICS . : { 1—96 ARITHMETIC AND ALGEBRA : : 5 . . I1—128 EXAMPLES OF THE PROCESSES . ; : : : 1—96 Chairman—The Right Hon. LO Capt. F. Beaufort, R.N., F.R.| COMMITTEE. Treasurer—WILLIAM TOOKE, Esq., M.P., F.R.S. W. Allen, Esq., F.R. & R.A.S./Sir Henry Ellis, Prin.Lib.Brit |Th. Hewitt Key, Esq., M.A. Mus. and R.A.S., Hydrographer to/John Elliotson, M.D., F.R.S. the Admiralty. G. Burrows, M.D. P. Stafford Carey, Esq., M.A. William Coulson, Esq. R. D. Craig, Esq. J. F. Daniell, Esq., F.R.S. J.F. Davis, Esq., F.R.S. Thomas Falconer, Esq. I. L. Goldsmid, Esq., F.R. and R.A.S. B. Gompertz, Esq., F.R. and R.AS G. B. Greenough, Esq., F.R. & | Ss a H. T. Dela Beche, Esq., F.R.S.'M. D. Hill, Esq. The Right Hon. Lord Denman.| Samuel Duckworth, Esq. The Right Rev. the Bishop of Durham, D.D. Rowland Hill, Esq., F.R.A.S. Kdwin Hill, Esq. The Rt. Hon. Sir J. C. Hob- house, Bart., M.P. Right Hon. Viscount Ebring-'David Jardine, Esq., A.M. ton, M.P. J. T. Leader, Esq., M.P. George C. Lewis, Esq., M.A. T. H. Lister, Esq. James Loch, Esq., M.P.. F.G.S: George Long, Esq., M.A. RD BROUGHAM, F.R.S., Member of the National Institute of France. Vice-Chairman—JOHN WOOD, Esq. The Rt, Hon. Sir H. Parnell, Bart., M.P. Dr. Roget, Sec. R.S.,F.R.A.S. Edward Romilly, Esq., M.A. Right Hon. Lord J. Russell, M.P. Sir M.A. Shee, P.R.A., F-R.S. J. W.Lubbock, Esq.,F.R.,R.A.'J. Abel Smith, Esq., M.P. and L.S.S. H. Malden, Esq., M.A. A. T. Malkin, Esq., M.A. James Manning, Esq. J.HermanMerivale,Esq., M.A. F.A.S James Mill, Esq. The Rt, Hon. Lord Nugent ‘Henry B. Ker, Esq. W.H. Ord, Esq., M.P. T.F.Ellis, Esq.,M.A.,F.R8.A.S. Rt. Hn.the Earl of Kerry,M.P.! LOCAL COMMITTEES. Alston, Staffordshire —Rev. J.| Chester.—Hayes Lyon, Esq. P. Jones. Anglesea.—Rev. E, Williams, Rev. W. Johnson. Mr. Miller. Ashburton—J. F. Kingston, Esq. Henry Potts, Esq. Chichester—F orbes,M.D.F.R.S. C. C. Dendy, Esq. Corfu—John Crawford, Esq, Mr. Plato Petrides. Barnstaple — — Bancraft, Esq'| Coventry.—Art. Gregory, Esq., William Gribble, Esq. Belfast—Dr. Drummond. Bilston.— Rev. W. Leigh. Birmingham.—JohnCorrie,Esq. F.RS., Chairman. Paul Moon James, Esq., Treasurer. Bridport.—Wm. Forster, Esq. James Williams, Esq. Denbigh—Jotun Madocks, Esq. Thos. Evans, sq. Derby—Joseph Strutt, Esq. Edward Strutt, Esq., M.P. Devonport and Stonehouse. John Gole, Esq, — Norman, Esq. Lieut-Col. C, Hamilton Smith, F.R.5. Bristol—J. N. Sanders, Esq.,|Dublin—T, Drummond, Esq., Chairman. J. Reynolds, Esq., Treas. J. B. Estlin, Esq., F.L.S, Sec. R.E,, F.R.A.S. Edinburgh—Sir Charles Bell, F.R.S.L and E. Etruria—Jos. Wedgwood, Esq. Caleutta—Lord Wm. Bentinck |#xeter—J. Tyrrell, Esq. Sir Edward Ryan. James Young, Esq. Cambridge—Rev. James Bow- stead, M.A. Rev. Prof. Henslow, M.A., F.L.S. & G.S. nd Leonard Jenyns, M.A., Rev. John Lodge, M. A. Rev. Geo. Peacock, M.A., F.R.S. & G.S. R. W. Rothman, Esq.,M.A. F.R.A.S., & G.S. Rev. Prof. Sedgwick, M. A., F.R.S. & G.S. Professor Smyth, M.A. Rev. C. Thirlwall, M.A. Canterbury—John Brent, Esq., Alderman. William Masters, Esq. Cardign—Rev. J. Blackwell, M.A Carlisle—Thos. Barnes, M.D., F.R.S.E. Carnarvon.—R. A. Poole, Esq William Roberts, Esq. John Milford, Esq.(Coaver.) Glasgow—\k. Finlay, Esq. Professor Mylne. Alexander Mc(Grigor, Esq. Charles ‘Tennant, Esq. , James Cowper, Esq, Glamorganshire— Dr. Malkin, Cowbridge. W. Williams, Esq. Aher- pergwm. Guernsey.—F. C. Lukis, Esq. Hull.—J.C. Parker, Esq. Keighley, Yorkshire —Rev. T, Dury, M.A, Launceston—Rev. J. Barfitt. Leamington Spa—Dr. Loudon, M.D Leeds—J. Marshall, Esq. Lenes—J. W. Woollgar, Esq. Limerick—Wm, O’Brien, Esq. Liverpool Local Association. W.W.Currie, Esq.Chairman. J. Mulleneux, Esq., reas. Rev. W. Shepherd. J. Ashton Yates, Esq. Ludlow—T. A. Knight, Esq., P.H.S. ae ascane Goolden, Esq., Maidstone.— Clement T. Smyth, Esq. John Case, Esq. Malmesbury.—B. C. Thomas, Esq. Manchester Local Association. G.W.Wood, Esq.,Chairman. Benj.Heywood, Esq., Treas. T. W. Winstanley, Esq., Hon. Sec. Sir G. Philips, Bart., M.P. Benjamin Gott, Esq. Masham—Reyv. George Wad- dington, M.A. Merthyr Tydvil—J. J. Guest, Esq., M.P. Min aer ce. G. Ball, sq. Songun H. Moggridge, Esq. Neath—John Rowland, Esq. Nencastle— Rev. W. Turner. T. Sopwith, Esq. F.G.8. Newport, Isle of Wight— Ab. Clarke, Esq. Rt. Hon. Earl Spencer. John Taylor, Esq., F.R.S. Dr. A. T. Thomson, F.L.S. H. Waymouth, Esq. J.Whishaw, Esq.,M.A., F.R.S John Wood, Esq. John Wrottesley, Esq.. M.A. F.R.A.S, J. A. Yates, Esq. E. Moore, M.D. F.L.S. Sec G.Wightwick, Esq, mn os, A. W. Davies, Rippon.—Rev. H. P. Hamilton, A.M., F.R.S. and G.S, Rev. P. Ewart, M.A. Ruthen.—Rev. the Warden of. Humphreys Jones, Esq. Ryde, Isle of Wight.— Sir Rd. Simeon, Bart., M.P. Shefjield—J. H. Abraham, Esq. Shepton Mallet.-— G. F. Burroughs, Esq. Shrensbury—R. A, Slaney, Esq., M.P. South Petherton—John Nicho- letts, Esq. St. Asaph.—Rev. Geo, Strong. Stockport—Henry Marsland, Esq., Treasurer. Henry Coppock, Esq., Sec Tavistock—Rev. W. Evans. John Rundle, Esq. Truro—Richard Taunton,M.D Henry Sewell Stokes, Esq. ae Wells.—Dr. Yeats, M.D. Uttoxeter—R. Blurton, Esq. T. Cooke, Jun., Esq. R. G. Kirkpatrick, Esq. Newport Pagnell—J Millar, Esq. Newtown, Montgomeryshire— William Pugh, Esq. Normich—Richard Bacon, Esq Orsett, Essex—Dr. Corbett, M.D. Warnick—Dr. Conolly, The Rev. William Field, (Leam.) Waterford—Sir John Newport, Bart. Wolverhampton — J. Esq. Worcester—Dr. Hastings, M.D. C. H. Hebb, Esq. Pearson, Oxford—Dr. Daubeny, F.R.S., Professor of Chemistry. Rev. Professor Powell. Rev. John Jordan, B.A. E. W. Head, Esq., M.A. Penang—Sir B. H. Malkin. Pesth, Hungary—Count Sze- chenyi Plymouth — H. Woollcombe, Esq., F.A.S., Chairman. Snow Harris, Esq., F.R.S. THOMAS COATES, Esq., Secretary, 59, Lincoln’s inn Fields. idiot Som 2 6g Edgworth, sq. J, E..Bowman,Esq., F.L.S. Treasurer. Major William Lloyd. Yarmouth—C. E,. Rumbold, Kisq., M.P. Dawson Turner, Esq. York—Rev. J. Kenrick, A.M, J. Phillips, Esq., F,R.S., F.GS. ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. Cuapter I. Introductory Remarks on the Nature and Objects of Mathematics. Tue object of this Treatise is—1. To point out to the student of Mathematics, who has not the advantage of a tutor, _ the course of study which it is most ad- visable that he should follow, the extent to which he should pursue one part of the science before he commences ano- ther, and to direct him as to the sort of applications which he should make. 2. To treat fully of the various points which inyolve difficulties and which are apt to be misunderstood by beginners, and to describe at length the nature without going into the routine of the operations which have been already dis- cussed in the Treatises of Arithmetic, Algebra, and Geometry, published hy this Society. No person commences the study of mathematics without soon discovering that it is of a very different nature from those to which he has been accustomed. The pursuits to which the mind is usually directed before entering on the sciences of algebra or geometry, are such as languages and history, &c. Of these, neither appears to have any affinity with mathematics ; yet, in order to see the difference which exists between these ‘studies, for instance, history and geo- metry, it will be useful to ask how we come by knowledge in each: suppose, for example, we feel certain of a fact related in history, such as the murder of Ceesar, whence did we derive the cer- tainty ? how came we to feel sure of the general truth of the circumstances of the narrative? The ready answer to this question will be, that we have not abso- lute certainty upon this point; but that _ we have the relation of historians, men of credit, who lived and published their accounts in the very time of which they write; that succeeding ages have re- ceived those accounts as true, and that succeeding historians have backed them with a mass of circumstantial evidence _ which makes it the most improbable thing in the world that the account, or any material part of it, should be false. This is perfectly correct, nor can there be the slightest objection to believing the whole narration upon such grounds; nay, our minds are so constituted, that, upon our knowledge of these arguments, we cannot help believing, in spite of our- selves. But this brings us to the point to which we wish to come; we believe that Caesar was assassinated by Brutus and his friends, not because there is any absurdity in supposing the contrary, since every one must allow that there is just a possibility that the event never happened: not because we can show that it must necessarily have been that, at a particular day, at a particular place, a successful adventurer must have been murdered in the manner described, but because our evidence of the fact is such, that, if we apply the notions of evidence which every-day experience justifies us in entertaining, we feel that the impro- bability of the contrary compels us to take refuge in the belief of the fact ; and, if we allow that there is still a possibility of its falsehood, it is because this sup- position does not involve absolute absur- dity, but only extreme improbability. In mathematics the case is wholly different. It is true that the facts as- serted in these sciences are of a nature totally distinct from those of history ; so much so, that a comparison of the evidence of the two may almost excite asmile. But if it be remembered that acute reasoners, in every branch of learning, have acknowledged the use, we might almost say the necessity, of a mathematical education, it must be ad- mitted that the points of connexion be- tween these pursuits and others are worth attending to. They are the more so, because there is a mistake into which several have fallen, and have deceived others, and perhaps themselves, by clothing some false reasoning in what they called a mathematical dress, ima- gining that, by the application of mathe- matical symbols to their subject, they secured mathematical erent This 2 STUDY OF could not have happened if they had possessed a knowledge of the bounds within which the empire of mathematics is contained. That empire is sufficiently wide, and might have been better known, had the time which has been wasted in aggressions upon the domains of others, been spent in exploring the immense tracts which are yet untrodden. We have said that the nature of ma- thematical demonstration is totally dif- ferent from all other, and the difference consists in this—that, instead of showing the contrary of the proposition asserted to be only improbable, it proves it at once to be absurd and impossible. This is done by showing that the contrary of the proposition which is asserted is in direct contradiction to some extremely’ evident fact, of the truth of which our eyes and hands convince us. In geo- metry, of the principles alluded to, those which are most commonly used are— I. If a magnitude be divided into parts, the whole is greater than either of those parts. II. Two straight lines cannot inclose a space. III. Through one point only one straight line can be drawn, which never meets another straight line, or which is parallel to it. It is on such principles as these that the whole of geometry is founded, and the demonstration of every proposition consists in proving the contrary of it to be inconsistent with one of these. Thus, in Euclid, Book I., Prop. 4, it is shown that two triangles which have two sides and the included angle respectively equal are equal in all respects, by proving that, if they are not equal, two straight lines will inclose a space, which is im- possible. In the Treatise on Geometry, Prop. 4, the same thing is proved in the same way, only the self-evident truth asserted differs in form from that of Euclid, and may be deduced from it, thus— Two straight lines which pass through the same two points must either inclose a space, or coincide and be one and the same line, but they cannot inclose a space, therefore they must coincide. Either of these propositions being granted, the other follows immediately; it is, there- fore, immaterial which of them we use. We shall return to this subject in treat- ing specially of the first principles of geometry. _ Such being the nature of mathema- tical demonstration, what we have before asserted is evident, that our assurance of a geometrical truth is of a nature wholly distinct from that which we can by any means obtain of a fact in history or an asserted truth of metaphysics. In reality, our senses are our first mathematical instructors ; they furnish us with notions which we cannot trace any further or re- present in any other way than by using single words, which every one under- stands. Of this nature are the ideas to which we attach the terms number, one, two, three, &c., point, straight line, sur- face ; all of which, let them be ever so much explained, can never be made any clearer than they are already to a child — of ten years old. But, besides this, our senses also furnish us with the means of reasoning on the things which we call by these names, in the shape of incon- trovertible propositions, such as have been already cited, on which, if any remark is made by the beginner in ma- thematics, it will probably be, that from such absurd truisms as “the whole is greater than its part,’ no useful result can possibly be derived, and that we might as well expect to make use of **two and two make four.” This obser- vation, which is common enough in the mouths of those who are commencing geometry, is the result of a little pride which does not quite like the humble operation of beginning at the beginning, and is rather shocked at being supposed. to want such elementary information. But it is wanted, nevertheless ; the low- est steps of a ladder are as useful as the highest. Now, the most common re- flexion on the nature of the propositions referred to will convince us of their truth, But they must be presented to the un- derstanding, and reflected on by it, since, simple as they are, it must be a mind of a very superior cast which could by itself embody these axioms, and proceed from them only one step in the road pointed out in any treatise on geometry. But, although there is no study which presents so simple a beginning as that of geometry, there is none in which dif- ficulties grow more rapidly as we pro- ceed, and what may appear at first rather paradoxical, the more acute the student the more serious will the impediments in the way of his progress appear. This necessarily follows in a science which consists of reasoning from the very com- mencement, for it is evident that every student will feel a claim to have his ob- jections answered, not by authority, but by argument, and that the intelligent MATHEMATICS. 3 student will perceive more readily than another the force’ of an objection and the obscurity arising from an unexplained difficulty, as the greater is the ordinary light the more will occasional darkness be felt. To remove some of these difficulties is the principal object of this Treatise. We shall now make a few remarks on the advantages to be derived from the study of mathematics, considered both as a discipline for the mind and a key to the attainment of other sciences. It is admitted by all that a finished or even a competent reasoner is not the work of nature alone; the experience of every day makes it evident that education de- velopes faculties which would otherwise never have manifested their existence. It is, therefore, as necessary to dearn to reason before we can expect to be able to reason, as it is to learn to swim or fence, in order to attain either of those arts. Now, something. must be rea- soned upon, it matters not much what it is, provided that it can be reasoned upon with certainty. The properties of mind or matter, or the study of languages, mathematics, or natural history, may be chosen for this purpose. Now, of all these, it is desirable to choose the one which admits of the reasoning being verified, that is, in which we can find out by other means, such as measure- ment and ocular demonstration of all sorts, whether the results are true or not. When the guiding property of the load- stone was first ascertained, and it was necessary to learn how to use this new discovery, and to find out how far it might be relied on, it would have been thought advisable to make many pas- sages between ports that were well known before attempting a voyage of discovery. So it is with our reasoning faculties : itis desirable that their powers should be exerted upon objects of such a nature, that we can tell by other means whether the results which we obtain are true or false, and this before it is safe to trust entirely toreason. Now the mathematics are peculiarly well adapted for this purpose, on the following grounds :— 1. Every term is distinctly explained, and has but one meaning, and it is rarely that two words are employed to mean the same thing. 2. The first principles are self-evident, and, though derived from observation, do not require more of it than has been made by children in general. 8 The demonstration is strictly logi- cal, taking nothing for granted except the self-evident first principles, resting nothing upon probability, and entirely independent of authority and opinion. 4, When the conclusion is attained by reasoning, its truth or falsehood can be ascertained, in geometry by actual measurement, in algebra by common arithmetical calculation. This gives confidence, and is absolutely necessary, if, as was said before, reason is not to be the instructor, but the pupil. 5. There are no words whose mean- ings are so much alike that the ideas which they stand for may be confounded. Between the meanings of terms there is no distinction, except a total distinction, and all adjectives and adverbs express- ing difference of degrees are avoided. Thus it may be necessary to say “A is greater than B ;” but it is entirely unim- portant whether A is very little or very much greater than B. Any proposition which includes the foregoing assertion will prove its conclusion generally, that is, for all cases in which A is greater than B, whether the difference be great or little. Locke mentions the dis- tinctness of mathematical terms, and says in illustration, ‘‘ The idea of two is as distinct from the idea of three as the magnitude of the whole earth is from that of a mite. This is not so in other simple modes, in which it is not so easy, nor perhaps possible for us to distinguish between two approaching ideas, which yet are really different; for who will undertake to find a difference between the white of this paper, and that of the next degree to it ?” These are the principal grounds cn which, in our opinion, the utility of ma- thematical studies may be shewn to rest, as a discipline for the reasoning powers. But the habits of mind which these stu- dies have a tendency to form are valua- ble in the highest degree. The most important of all is the power of concen- trating the ideas which a successful study of them increases where it did exist, and creates where it didnot. A difficult position, or a new method of passing from one proposition to another, arrests all the attention, and forces the united faculties to use their utmost exer- tions. The habit of mind thus formed soon extends itself to other pursuits, and is beneficially felt in all the business of life. As a key to the attainment of other sciences, the use of the mathematics is too well known to make - peconhy * 4 STUDY OF that we should dwell on this topic. In fact, there is not in this country any dis- position to undervalue them as regards the utility of their applications. But though they are now generally consi- dered as a part, and a necessary one, of a liberal education, the views which are still taken of them as a part of education bya large proportion of the community are still very confined. The elements of mathematics usually taught are contained in the sciences of arithmetic, algebra, geometry, and tri- gonometry. We have used these four divisions because they are generally adopted, though, in fact, algebra and geometry are the only two of them which are really distinct. Of these we shall commence with arithmetic, and take the others in succession in the order in which we have arranged them. CuaPTer II. On Arithmetical Notation. Tue first ideas of arithmetic, as well as those of other sciences, are derived from early observation. How they come into the mind it is unnecessary to inquire ; nor ‘is it possible to define what we mean by number and quantity. They are terms so simple, that is, the ideas which they stand for are so completely the first ideas of our mind, that it is im- possible to find others more simple, by which we may explain them. This is what is meant by defining aterm; and here we may say a few words on defini- tions in general, which will apply equally to all sciences. Definition is the explaining a term by means of others, which are more easily understood, and thereby fixing its mean- ing, so that it may be distinctly seen what it does imply, as well as what it does not. Great care must be taken that the definition itself is not a tacit assumption of some fact or other which ought to be proved. Thus, when it is said that a square is “a four-sided figure, all whose sides are equal, and all whose angles are right angles,” though no more is said than is true of a square, yet more is said than is necessary to define it, because it can be proved that if a four-sided figure have all its sides equal, and one only of its angles a right angle, all the other angles must be richt angles also. Therefore, in making the above definition, we do, in fact, affirm that which ought to be proved. Again, the above definition, though redundant in one point, is, strictly speaking, defec- tive in another, for it omits to state whether the sides of the figure are straight lines/or curves. It should be, ‘‘a square is a four-sided rectilinear figure, all of whose sides are equal, and one of whose angles is a right angle.” As the mathematical sciences owe much, if not all, of the superiority of their demonstrations to the precision with which the terms are defined, it is most essential that the beginner should see clearly in what a good definition consists. We have seen that there are terms which cannot be defined, such as number and quantity. An attempt ata definition would only throw a difficulty in the student’s way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatises on that subject. A point is defined to be that ‘ which has no parts, and which has no magnitude ;”’ a straight line is that which “lies evenly between its extreme points.” Now, let any one ask himself whether he could have guessed what was meant, if, before he began geometry, any one had talked to him of ‘that which has no parts and which has no magnitude,’ and “the line which lies evenly between its extreme points,” unless he had at the same time mentioned the words ‘“ point’ and ** straight line,” which would have re- moved the difficulty ? In this case the explanation is a great deal harder than the term to be explained, which must always happen whenever we are guilty of the absurdity of attempting to make the simplest ideas yet more simple. A knowledge of our method of reckon- ing, and of writing down numbers, is taught so early, that the method by which we began is hardly recollected. Few, therefore, reflect upon the very commencement of arithmetic, or upon the simplicity and elegance with which calculations are conducted. We find the method of reckoning by ten in our hands, we hardly know how, and we conclude, so natural and obvious does it seem, that it came with our language, and is a part of it; and that we are not much indebted to instruction for so sim- ple a gift. It has been well observed, that if the whole earth spoke the same language, we should think that the name of any object was not a mere sign chosen to represent it, but was a sound which had some real connexion with the thing ; - and that,we should laugh at, and per- MATHEMATICS. 5 haps persecute, any one who asserted that any other sound would do as well if we chose to think so. We cannot fall into this error, because, as it is, we happen to know that what we call by the sound “horse,” the Romans distin- guished as well by that of ‘‘ equas,” but we commit a similar mistake with regard to our system of numeration, because at present it happens to be received by all civilized nations, and we do not reflect on what was done formerly by almost all the world, and is done still by savages. The following considerations will, per- haps, put this matter on a right footing, and shew that in our ideas of arithmetic we have not altogether rid ourselves of the tendency to attach ideas of mysticism to numbers which has prevailed so ex- tensively in all times. We know that we have nine signs to stand for the first nine numbers, and one for nothing, or zero. Also, that to represent ten we do not use a new sign, but combine two of the others, and de- note it by 10, eleven by 11, and so on. But why was the number ¢en chosen as the limit of our separate symbols—why not nine, eight, or eleven? If we recol- lect how apt we are to count on the fin- gers, we shall be at no loss to see the reason. We can imagine our system of numeration formed thus :—A man pro- ceeds to count a number, and to help the memory he puts a finger on the table for each one which he counts. He can thus go as far as ten, after which he must begin again, and by reckoning the fingers a second time he will have counted twenty, and so on. But this is not,enough; he must also reckon the number of times which he has done this, and as by counting on the fingers he has divided the things which he is counting into lots of ten each, he may consider each lot as a unit of its kind, just as we say anumber of sheepis ove flock, twenty shillings are one pound. Call each lot aiten. In this way he can count a ten of tens, which he may call a hundred, a ten of hundreds, or a thousand, and so on. The process of reckoning would then be as follows :—Suppose, to choose an example, a number of faggots is to be counted. They are first tied up in bun- dles of ten each, until there are not so many as ten left. Suppose there are seven over. We then count the bundles of ten as we counted the single faggots, and tie them up also by tens, forming new bun- dles of one hundred each with some bun- dies of tenremaining. Let these last be six in number. We then tie up the bun- dles of hundreds by tens, making bun- dles of thousands, and find that there are five bundles of hundreds remaining. Suppose that on attempting to tie up the thousands by tens, we find there are not so many as ten, but only four. The number of faggots is then 4 thousands, 5 hundreds, 6 tens, and 7. The next question is, how shall we re- present this number in a short and con- venient manner ? It is plain that the way to do this is a matter of choice. Suppose, then, that we distinguish the tens by marking their number with one accent, the hundreds with two accents, and the thousands with three. We may then represent this number in any of the following ways :— 76/54’, 6/75/'4!!’, 6/4/57, 4156/7, the whole number of ways being 24, But this is more than we want; one certain method of repre- senting a number is sufficient. The most natural way is to place them in order of magnitude, either puttmg the largest collection first or the smallest ; thus 4/"5/6'7, or 76'54'", Of these we choose the first. In writing down numbers in this way it will soon be apparent that the accents are unnecessary. Since the singly ac- cented figure will always be the second from the right, and so on, the place of each number will point out what accents to write over it, and we may therefore consider each figure as deriving a value from the place in which it stands. But here this difficulty occurs. How are we to represent the numbers 3//3/, and 4''2'7 without accents ? If we write them thus, 33 and 427, they will be mistaken for 3’3 and 4//2'7, This difficulty will be obviated by placing cyphers so as to bring each number into the place allotted to the sort of collection which it repre- sents; thus, since the trebly accented letters, or thousands, are in the fourth place from the right, and the singly accented letters in the second, the first number may be written 3030, and the second 4027. The cypher, which plays so important a part in arithmetic that it was anciently called the art of cypher, or cyphering, does not stand for any number initself, but is merely employed, like blank types in printing, to keep other signs in those places which they must occupy in order to be read rightly. We may now ask what would have been the case if, instead of ten fingers, men had had more or less. For example, by what signs would 4567 have been repre- ‘ STUDY OF sented, if man had nine fingers instead of ten? We may presume that the method would have been the same with the number nine represented by 10 instead of ten, and the omission of the symbol 9, Suppose this number of faggots is to be counted by nines. Tie them up in bun- dles of nine, and we shall find 4 fag- gots remaining. Tie these bundles again in bundles of nine, each of which will, therefore, contain eighty-one, and there will be 3 bundles remaining. These tied up in the same way into bundles of nine, each of which contains seven hun- dred and twenty-nine, will leave 2 odd OC Naas 4 5 Counting by tens....1 2 Counting by nines...1 2 URS Waa ee ee Rig ie Bod ih ett V6 VR a O° aly gfe 4319 We will now write, in the common way, in the tens’ system, the process which we went through in order to find 9)4567 9) 507 — rem. 4. 9) 56 — rem. 3, 9) 6 — rem. 2. 0 — rem. 6. The processes of arithmetic are the same in principle whatever system of numeration is used. To show this, we subjoin a question in each of the first four rules, worked both in the common 40 50 60 70 8&0 4!4 7°55 6! 77° BS TT IO TNT bundles, and, as there will be only six of them, the process cannot be carried any further. If, then, we represent, by 1’, a bundle of nine, or a nine, by 1/’ a nine of nines, and so on, the number which we write 4567, must be written 6/// 2! 3! 4, In order to avoid confusion, we will suffer the accents to remain over all numbers which are not reckoned in tens, while those which are so reckoned shall be written in the common way. The following is a comparison of the way in which numbers in the common system are written, and in the one which we have just explained :— Got Beco 1 BL he ale oo Be FO SAO TT Pore Thais 90 100 how to represent the number 4567 in that of the nines, thus :— Representation required, 6!” 2" 3/4, system, and in that ofthe nines. There is this difference, that, in the first, the tens must be carried, and in the second the nines. ADDITION, 636 gualg 987 W316 403 Alara 2026 Qi ail Oly SUBTRACTION, 1384 1! 8 0' 7 797 VO 715 587 712! 2 MULTIPLICATION, 297 3’ 6' 0 136 17 6/1 1782 3.60 891 240 0 297 36 0 40392 gilt V3! 6 0. MATHEMATICS. 7 : DIVISION. 633) 79125 (125 77/3) 1% Biv Oli git 716* (1/'4'8 633 Ph aigee a 1582 A A 1266 By 423 3165 6 8 4 6 3165 6 8 4 6 0 0 The student should accustom himself to work questions in different systems of numeration, which will give him a clearer insight into the nature of arithmetical processes than he could obtain by any other method. When he uses a system in which numbers are counted by a num- ber greater than ten, he will want some new symbols for figures. For example, in the duodecimal system, where twelve is the number of figures supposed, twelve will be represented by 1/0; there must, therefore, be a distinct sign for ten and eleven, a nine and six reversed, thus, g and 3, might be used for these. Cuaprer III. Elementary Rules of Arithmetic. As soon as the beginner has mastered the notation of arithmetic, he may be made acquainted with the meaning of the algebraical signs +, -, X, =, and also with that for division, or the com- mon way of representing a fraction. There is no difficulty in these signs or in their use. Five minutes’ consideration will make the symbol 5+3 present as clear an idea as the words ‘5 added to 3.” The reason why they usually cause so much embarrassment is, that they are generally deferred until the student commences algebra, when he is often introduced at the same time to the representation of numbers by letters, the distinction of known and unknown quantities, the signs of which we have been speaking, and the use of figures as exponents of letters. Either of these four things is quite sufficient at a time, and there is no time more favourable for beginning to make use of the signs of operation than when the habit of per- forming the operations commences. The beginner should exercise himself in * To avoid too great a number of accents, Roman numerals are put instead of them ; also, to avoid confusion, the accents are omitted after the first line. putting the simplest truths of arithmetic in this new shape, and should write such sentences as the following frequently :— 2+7=9, 6-4=2, 14+8+4—-6=4+2-+1, 2xX24+12x12=14x104+2x2xz. These will accustom him to the meaning of the signs, just as he was accustomed to the formation of letters by writing copies. As he proceeds through the rules of arithmetic he should take care never to omit connecting each operation with its sign, and should avoid con- founding operations together, and consi- dering them as the same, because they produce the same result. Thus, 4x7 does not denote the same operation as 7x4, though the result of both is 28. The first is four multiplied by seven, four taken seven times; the second is seven multiplied by four, seven taken four times; and that 4x7=7x4 is a proposition to be proved, not to be taken for granted. Again, 1 x 4 and 4 are marks of distinct operations, though their result is the same, :as we shall show in treating of fractions. The examples which a beginner should choose for practice should be simple, and should not contain very large numbers. The powers of the mind cannot be di- rected to two things at once: if the complexity of the numbers used requires all the student’s attention, he cannot observe the principle of the rule which he is following. Now, at the commence- ment of his career, a principle is not received and understood by the student as quickly as it is explained by the in- structor. Hedoes not, and cannot, ge- neralize at all; he must be taught to do so; and he cannot learn that a particular fact holds good for all numbers unless by having it shown that it holds good for some numbers, and that for those some numbers he may substitute others, and use the same demonstration. Until 8 STUDY OF he can do this himself he does not un- derstand the principle, and he can never do this except by seeing the rule ex- plained, and trying it himself on small numbers. He may, indeed, and will, believe it on the word of his instructor, but this disposition is to be checked. He must be told that, whatever is not gained by his own thought is not gained to any purpose; that the mathematics are put in his way purposely because they are the only sciences in which he must not trust the authority of any one. The superintendence of these efforts is the real business of an instructor in arithmetic. The merely showing the student a rule by which he is to work, and comparing his answer with a key to the book, printed for the preceptor’s private use, to save the trouble which he ought to bestow upon his pupil, is not teaching arithmetic any more than pre- senting him with a grammar and die- tionary is teaching him Latin. When the principle of each rule has been well established by showing its application to some simple examples (and the number of these requisite will vary with the in- tellect of the student), he may then pro- ceed to more complicated cases, in order to acquire facility in computation. The four first rules may be studied in this way, and these will throw the greatest light on those which succeed. The student must observe that all operations in arithmetic may be resolved into addition and subtraction ; that these additions and subtractions might be made with counters; so that the whole of the rules consist of processes in- tended to shorten and simplify that which would otherwise be long and complex. For example, multiplication is continued addition of the same num- ber to itself—twelve times sevenis twelve sevens added together. Division is a continued subtraction of one number from another; the division of 129 by 3 is a continued subtraction of 3 from 129, in order to see how many threes it con- tains. All other operations are com- posed of these four, and are, therefore, the result of additions and subtractions only. The following principles, which occur so continually in mathematical opera- tions that we are, at length, hardly sen- sible of their presence, are the foun- dation of the arithmetical rules :-— I. We do not alter the sum of two numbers by taking away any part of the first, if we annex that part to the second. This may be expressed by signs, ina particular instance, thus :— (20 — 6) + (832+6) = 20+32. IJ. We do not alter the difference of two numbers by increasing or diminish- ing one of them, provided we increase or diminish the other as much. This may be expressed thus, in one instance :— (4547) — (22+7)=45 — 29, (45 —8) — (22 — 8) =45 — 22. II]. If we wish to multiply one num- ber by another, for example 156 by 29, we may break up 156 into any number of parts, multiply each of these parts by 29, and add the results. For example, 156 is made up of 100, 50, and 6. Then 156 X 29=100 X29+50xX29+6 x 29, IV. The same thing may be done with the multipher instead of the multi- plicand. Thus, 29 is made up of 18, 6, and 5. Then 156 X29=156 X18+156X6+156 x5. V. If any two or more numbers be multiplied together, it is indifferent in what order they are multiplied, the result is the same. Thus, 10x6x4xX3 =38x10x4ax6=6 X10 4x3, &c. VI. In dividing one number by ano- ther, for example 156 by 12, we may break up the dividend, and divide each of its parts by the divisor, and then add the results. We may part 156 into 72, 60, and 24; this is expressed thus :— 156 72, 60 24 Ag eo te ee The same thing cannot be done with the divisor. It is not true that 156 156 156 156 12 4 Be tts te The student should discover the reason for himself. A prime number is one which is not divisible by any other number except 1. When the process of division can be performed, it can be ascertained whether a given number is divisible by any other number, that is, whether it is prime or not. This can be done by dividing it by all the numbers which are less than its half, since it is evident that it cannot be divided into a number of parts, each of which is greater than its half. This process would be laborious when the given number is large; still it may be done, and by this means the number itself may be reduced to tts prime fac- MATHEMATICS. 9 tors,* as it is called, that is, it may either be shewn to be a prime number itself or made up by multiplying several prime numbers together. Thus, 306 is 34x9, or 2X17X9, or 2X17x3X3, and has for its prime factors 2, 17, and 3, the latter of which is repeated twice in its formation. When this has been done with two numbers, we can then see whether they have any factors in com- mon, and, if that be the case, we can then find what is called their greatest common measure or divisor, that is, the number made by multiplying all their common factors. It is an evident truth that, if a number can be divided by the product of two others, it can be divided by each of them. If a number can be parted into an exact number of twelves, it can be parted also into a number of sixes, twos, or fours. It is also true that, if a number can be divided by any other number, and the quotient can then be divided by a third number, the original number can be divided by the product of the other two. Thus, 144 is divisible by 2; the quotient, 72, is divi- sible by 6; and the original number is divisible by 6 x2 or 12. It is also true that, if two numbers are prime, their product is divisible by no numbers ex- cept themselves. Thus, 17 X11 is divi- sible by no numbers except 17 and 11. Though this is a simple proposition, its proof is not so, and cannot be given to the beginner. From these things it follows that the greatest common mea- sure of two numbers (measure being an old word for divisor) is the product of all the prime factors which the two possess in common. For example, the num- bers 90 and 100, which, when reduced to their prime factors, are 2 x 5 X 3x 3 and 2 x 2x5 x 5, have the common factors 2 and 5, and are divisible by 2x 5,or10. The quotients are 3 x 3 and 2 x 5, or 9 and 10, which have no common factor remaining, and 2 x 5, or 10, is the greatest common measure of 90 and 100. The same may be shewn in the case of any other numbers. But the method we have mentioned of resoly- ing numbers into their prime factors, being troublesome to apply when the numbers are large, is usually abandoned for another. It happens frequently that a method simple in principle is laborious in practice, and the contrary. When one number is divided by ano- ther, and its quotient and remainder * The factors of a number are those numbers by the multiplication of which it is made. obtained, the dividend may be recovered again by multiplying the quotient and divisor together, and adding the remain- der to the product. Thus 171 divided by 27 gives a quotient 6 and a remain- der 9, and 171 is made by multiplying 27 by 6, and adding 9 to the product. That is, 171 = 27 x6-+9. Now, from this equation it is easy to shew that every number which divides 171 and 27 also divides 9, that is, every common measure of 171 and 27 is also a common measure of 27 and 9. We can also shew that 27 and 9 have no common measures which are not common to 171 and 27. Therefore, the common measures of 171 and 27 are those, and no others, which are common to 27 and 9; the greatest com- mon measure of each pair must, there- fore, be the same, that is, the greatest common measure of a divisor and divi- dend is also the greatest common mea- sure of the remainder and divisor. Now take the common process for finding the greatest common measure of two num- bers ; for example, 360 and 420, which is as follows, and abbreviate the words greatest common measure into their ini- tials g. c. m.:— 360 ) 420 (1 360 60 ) 360 (6 360 0 From the theorem above enunciated it appears that g.c.m. of 420 and 360 is g.c. m. of 60 and 360; g.c.m. of 60 and 360 is 60; because 60 divides both 60 and 360, and no number can have a greater measure than itself. Thus may be seen the rea- son of the common rule for finding the greatest common measure of two num- bers. Every number which can be di- vided by another without remainder is called a multiple of it. Thus, 12, 18, and 42 are multiples of 6, and the last is acommon multiple of 6 and 7, because it is divisible both by 6 and 7. The only things which it is necessary to ob- serve on this subject are, Ist, that the product of two numbers is a common multiple of both; 2d, that when the two numbers haveacommon measure greater than 1, there is acommon multiple less than their product; 3d, that when they have no common measure except 1, the 10 least common multiple is their product. The first of these is evident; the second will appear from an example. Take 10 and 8, which have the common mea- sure 2, since the first is 2 x 5 and the second 2 x4. The product is 2x2 x4x 5, but 2x45 is also a common multiple, since it is divisible by 2 x 4, or 8, and by 2x 5, or 10. To find this common multiple we must, therefore, divide the product by the greatest common mea- sure. The third principle cannot be proved in an elementary way, but the student may convince himself of it by any number of examples. He will not, for instance, be able to find a common multiple of 8 and 7 less than 8 x 7, or 56. CHAPTER IV. Arithmetical Fractions. WueEn the student has perfected himself in the four rules, together with that for finding the greatest common measure, he should proceed at once to the subject of fractions. This part of arithmetic is usually supposed to present extraordi- nary difficulties; whereas, the fact is that there is nothing in fractions so diffi- cult, either in principle or practice, as the rule for finding the greatest common measure. We would recommend the student not to attend to the distinctions of proper and improper, pure or mixed fractions, &e. as there is no distinction whatever in the rules, which are common to all these fractions. - When one number, as 56, is to be divided by another, as 8, the process is written thus:—°,. By this we mean that 56 is to be divided into 8 equal parts, and one of these parts is called the quo- tient. In this case the quotient is 7. But it is equally possible to divide 57 into 8 equal parts; for example, we candivide 57 feet into 8 equal parts, but the eighth part of 57 feet will not bean exact num- ber of feet, since 57 does not contain an exact number of eights; a part of a foot will be contained in the quotient 57, and this quotient is therefore called a frac- tion, or broken number. If we divide 57 into 56 and 1, and take the eighth part of each of these, whose sum will give the eighth part of the whole, the eighth of 56 feet is 7 feet; the eighth of 1 foot is a fraction, which we write i, and 57 is 7-4-1, which is usually written 73. Both of these quantities 57, and 72, are called fractions; the only difference is that, in the second, that part of the STUDY OF quotient which is’a whole number is separated from the part which is less than any whole number. There are two ways in which a frac- tion may be considered. Let us take, for example, 5, This means that 5 is to be divided into 8 parts, and 2 stands for one of these parts. The same length will be obtained if we divide 1 into 8 parts, and take 5 of them, or find 1 x 5, To prove this let each of the lines drawn below represent 1 of an inch; repeat 4 five times, and repeat the same line eight times, Sete s= Seeeroe cee cee> Peeee ke Se Se ee In each column is Ath of an inch repeated 8 times; that is one inch. There are, then, 5 inches in all, since there are five columns. But since there are 8 lines, each line is the eighth of 5 inches, or 3, but each lineis also 4th of an inch repeated 5 times, or 4x5. Therefore, 5 =4%X 5; that is, in order to find 3 inches, we may either divide Jive inches into 8 parts, and take one of them, or divide one inch into 8 parts, and take five of them. The symbol 5 is made to stand for both these opera- tions, since they lead to the same result. The most important property of a fraction is, that if both its numerator and denominator are multiplied by the same number, the value of the fraction is not altered; that is, 2 is the same as 42, or each part is the same when we divide 12 inches into 20 parts, as when we divide 3 inches into 5 parts. Again, we get the same length by dividing 1 inch into 20 parts, and taking 12 of them, which we get by dividing 1 inch into 5 parts and taking 3 of them. This hardly needs demonstration. Taking 12 out of 20 is taking 3 out of 5, since for every 3 which 12 contains, there is a 5 con- tained in 20. Every fraction, therefore, admits of innumerable alterations in its form, without any alteration in its value. Thus, } =2=3=4 fy Se. 5 = > = 6 = 8 tr eH op &e. On the same principle it is shewn that MATHEMATICS. the terms of a fraction may be divided by any number without any alteration of its value. There will now be no difficulty in reducing fractions to a common deno- minator, in reducing a fraction to its lowest terms ; neither in adding nor sub- tracting fractions, for all of which the rules are given in every book of arith- metic. We now come to arule which presents more peculiar difficulties in point of principle than any at which we have yet arrived, If we could at once take the most general view of numbers, and give the beginner the extended notions which he may afterwards attain, the mathema- tics would present comparatively few impediments. But the constitution of our minds will not permit this. It is by coliecting facts and principles, one by one, and thus only, that we arrive at what are called general notions; and we afterwards make comparisons of the facts which we have acquired, and discover analogies and resemblances which, while they bind together the fabric of our knowledge, point out methods of in- creasing its extent and beauty. In the limited view which we first take of the operations which we are performing, the names which we give are necessarily confined and partial; but when, after additional study and reflection, we recur to our former notions, we soon discover processes so resembling one another, and different rules so linked together, that we feel it would destroy the symme- try of our language if we were to call them by different names. We are then induced to extend the meaning of our terms, so as to make tworules into one. Also, suppose that when we have disco- vered and applied a rule, and given the process which it teaches a particular name, we find that this process is only a part of one more general, which applies to all cases contained under the first, and to others besides. We have only the alternative of inventing a new name, or of extending the meaning of the for- mer one so as to merge the particular process in the more general one of which it is a part. Of this we can give an instance. We began with reasoning upon simple numbers, such as 1, 2, 3, 20, &c. We afterwards divided these into parts, of which we took some num- ber, and which we galled fractions, such as 2, 7,1, &c. Now thereis no num- ber which may not be considered as a fraction in as many different ways as we 11 please. Thus 7 is 14 oy 21, &e.; 12 is 194 13. &e ; ’ at. Ail tele tion is, then, one which includes all our former ideas of number, ‘and others be- sides, It is then customary to represent by the word number, not only our first notion of it, but also the extended one, of which the first is only a part. Those to which our first notions applied we call whole numbers, the others fractional numbers, but still the name number is applied both to 2 and 4, to3 and 3, The rules of which we have: spoken is another instance, It is called the multiplication of fractional numbers. Now, if we return to our meaning of the word multiplication, we shall find that the multiplication of one fraction b another appears an absurdity. We mu tiply a number by taking it several times and adding these together. What, then, is meant by multiplying by a fraction ? Still, a rule has been found which, in applying mathematics, it is necessary to use for fractions, in all cases where mul- tiplication would have been used had they been whole numbers. Of this we shall now give a simple example. Take an oblong figure (which is called a rectangle in geometry), suchas ABC D, and find the magnitudes of the sides AB and BC ininches, Draw the line | PN TO See AP NOE Lae si AOR oY Be Ee eR OPRE SES & (Mr lathe etd ih stb soi a aae Len wba EF equal in length to one inch, and the square G, each of whose sides is one inch. Ifthe lines AB, and BC contain an exact number of inches, the rectan- gle ABCD contains an exact number of squares, each equal to G, and the number k ——F of squares contained is found by multiply- ing the number of inches in A B by the G | number of inches in BC. In the present case the number of squares is 3 X 4, or 12. Now, suppose another rectangle A’ B/ C’ D’, of which neither of the sides is an exact number of inches; suppose, for example, that Our new notion of frac- 12 A’ B’ is 2 of an inch, B’ C’ and that B/C! is % of aninch. We may | | shew, by reasoning, that we can find how Al. D’ much A’ B’C’D’ is of G by forming a fraction which has the product of the numerators of 2 and 5 for its numerator, and the product of their denominators for its denominator ; that is, that A’ B/C’ D/ contains 3% of G. Here then appears a connexion between the multiplication of whole numbers, and the formation of a fraction whose numerator is the pro- duct of two numerators, and its denomi- nator the product of the corresponding denominators. These operations will always come together, . that is whenever a question occurs in which, when whole numbers are given, those numbers are to be multiplied together ; when fractional numbers are given, it will be necessary, in the same case, to multiply the nume- rator by the numerator, and the denomi- nator by the denominator, and form the result into a fraction, as above. This would lead us to suspect some connexion between these two operations, and we shall accordingly find that when whole numbers are formed into frac- tions, they may be multiplied together by this ivery rule. Take, for example, the numbers 3-and 4, whose product is 12. The first may be written as +, and the second as 8, Form a fraction from the product of the numerators and denominators of these, which will be 1%, which is 12, the product of 3 and 4. From these considerations it is cus- tomary to call the fraction which is pro- duced from two others in the manner above stated, the product of those two fractions, and the process of finding the third fraction, multiplication. We shall always find the first meaning of the word multiplication included in the second, in all cases in which the quanti- ties represented as fractions are really whole numbers. The mathematics are not the only branches of knowledge in which it is customary to extend the meaning of established terms. When- ever we pass from that which is simple to that which is complex, we shall see the necessity of carrying our terms with us, and enlarging their meaning, as we enlarge our own ideas. This is the only method of forming a language which STUDY OF shall approach in any degree towards perfection ; and more depends upon a well-constructed language in mathema- tics than in anything else. It isnot that an imperfect language would deprive us of the means of demonstration, or cramp the powers of reasoning. The proposi- tions of Euclid upon numbers are as rationally established as any others, although his terms are deficient in ana- logy, and his notation infinitely inferior to that which we use. Itis the progress of discovery which is checked by terms constructed so as to conceal resem-~ blances which exist, and to prevent one result from pointing out another. The higher branches of mathematics date the progress which they have made in the last century and a half, from the time when the genius of Newton, Leibnitz, Descartes, and Harict turned the atten- tion of the scientific world to the imper- fect mechanism of the science. A slight and almost a casual improvement, made by Hariot in algebraical language, has been the foundation of most important branches of the science*. The subject of the last articles is of very great import- ance, and will often recur to us in explain- ing the difficulties of algebraical notation. The multiplication of 5 by 3 is equi- 2 ~ valent to dividing 3 into 2 parts, and taking three such parts. Because 5. being the same as 19, or 1 divided into 12 parts and 10 of them taken, the half of 49 is 5 of those parts, or -5.. Three 13° times this quantity will be 15 of those parts, or 15, which is by our rule the same as what we have called, > multi- plied by 3. But the same result arises from multiplying 3 by 3, or dividing 3 into 6 parts and taking 5 of them. Therefore, we find that 3 multiplied by 3 is the same as 5 multiplied by 3,or 3 = 5 gx f= 5x 3, ANG usually considered as requiring no proof, because it is received very early on the authority of a rule in the elements of arithmetic. But it is not self-evident, for the truth of which we appeal to the beginner himself, and ask him whether he would have seen at once that 5 This proposition is of an apple divided into 2 parts and 3 of ROCCE ARTE LOY Mer he * The mathematician will be aware that I allude to writing an equation in the form “2 + ax— b = 0; instead of Ve + Gx =. MATHEMATICS. them taken, is the same as 3 ofan apple, or one apple and a-half divided into six parts and 5 of them taken. An extension of the same sort is made of the term division. In dividing one whole number by another, for example, 12 by 2, we endeavour to find how many twos must be added together to make 12. In passing from a problem which con- tains these whole numbers to one which contains fractional quantities, for exam- ple # and 2, it will be observed that in place of finding how many twos make 12, we shall have to find into how many parts 2 must be divided, and how many of them must be taken, so as to give #. If we reduce these fractions to a com- mon denominator, in which case they will be 15 and 8; and if we divide the second into 8 equal parts, each of which will be .1., and take 15 of these parts, BO? we shall get 15, or ?, The fraction whose numerator is 15, and whose de- nominator is 8, or 15, will in these pro- blems take the place of the quotient of the two whole numbers. In the same manner as before, it may be shewn that this process is equivalent to the division of one whole number by another, when- ever the fractions are really whole num- bers; for example, 3 is 1%, and 15 is as fa If this process be applied to 3; and #2; the result is am which is 5, or the same as 15 divided by 3. This pro- cess is then, by extension, called division: 15 is called the quotient of # divided by 2, and is found by multiplying the numerator of the first by the denomi- nator of the second for the numerator of the result, and the denominator of the first by the numerator of the second for the denominator of the result. That this process does give the same result as ordinary division in all cases where ordi- nary division is applicable, we can easily shew from any two whole numbers, for example, 12 and 2, whose quotient is 6. Now 12 is a0) and 2 is og and the rule for what we have called division of frac- tions will give as the quotient 180, which is 6. In all fractional investigations, when the beginner meets with a difficulty, he should accustom himself to leave the notation of fractions, and betake him- 13 self to their original definition. He should recollect that 5 is 1 divided into 6 parts and five of them taken, or the sixth part of 5, and he should reason upon these suppositions, neglecting all rules until he has established them in his own mind by reflection on particular instances. These instances should not contain large numbers, and it will per- haps assist him if he reasons on some given unit, for example a foot. Let AB be one foot, and divide it into any number of equal parts (7, for example) by the points C,D,E, F,G, and H. Agr goa wis tort baat Cc D E F G H He must then recollect that each of these parts is + of a foot; that any two of them together are 2 ofa foot; any 3, 3, and soon. He should then accustom himself, without a rule, to solve such questions as the following, by observa- tion of the figure, dividing each part into several equal parts, if necessary ; and he may be well assured that he does not understand the nature of fractions until such questions are easy to him. What is 1 of 2 of a foot? What is 2 of 1 of 3 of a foot? Into how many parts must 3 of a foot be divided, and how many of them must be taken to produce 14 of a foot? Whatis 1+ 14 of a foot? and so on. CHAPTER V. Decimal Fractions. Ir is a disadvantage attending rules received without a knowledge of princi- ples, that a mere difference of language is enough to create a notion in the mind of a student that he is upon a totally different subject. Very few beginners see that in following the rule usually called practice, they are working the same questions as were proposed in compound multiplication ;—that the rule of three is only an application of the doctrine of fractions; that the rules known by the name of commission, bro- kerage, interest, &c., are the same, and so on. No instance, however, is more conspicuous than that of decimal frac- tions, which are made to form a branch of arithmetic as distinct from ordinary or vulgar fractions as any two parts of the subject whatever. Nevertheless, 14 there is no single rule in the one which is not substantially the same as the rule corresponding in the other, the difference consisting altogether in a different way of writing the fractions. The beginner will observe that throughout the subject it is continually necessary to reduce fractions to a common denominator: he will see, therefore, the advantage of always using either the same denomina- tor, or a set of denominators, so closely connected as to be very easily reducible to one another. Now of all numbers which can be, chosen the most easily manageable are 10, 100, 1000, &c., which are called decimal numbers on account of their connexion with the number ten. All fractions, such as foo Too S222, which have a de- cimal number for the denominator, are called decimal fractions. Now a deno- minator of this sort is known whenever the number of cyphers in it are known ; thus a decimal number with 4 cyphers can only be 10,000, or ten thousand. We need not, therefore, write the deno- minator, provided, in its stead, we put some mark ‘upon the numerator, by which we may know the number of cyphers in the denominator. This mark is for our own selection. The method which is followed is to point off from the numerator as many /igwres as there are cyphers in the denominator. Thus 1334 is represented by 17.334; +°%% thus, .229. We might, had we so pleased, have represented them thus, 173348, 229%; or thus, 17334,, 229,, or in any way by which we might choose to agree to recollect that the denominator 26 3 21734 +-l0+7+= er 10000 We see, then, that in the fraction 217.3426 the first figure 2 counts two hundred ; the second figure, 1, ten, and the third 7 units. It appears, then, that all figures on the left of the decimal point are reckoned as ordinary numbers. But on the right of that point we find the figure 3, which counts for 3594 which counts;4,; 2, or 2; and 6, Orz5855- It appears, therefore, that numbers on the right of the decimal point decrease as they move towards the right, each number being one-tenth of STUDY OF is 1 followed by 3 cyphers. In the com- mon method this difficulty occurs imme- ~ diately. What shall be done when there are not as many figures in the nume- rator as there are cyphers in the deno- minator? How shall we _ represent We must here extend our language a little, and imagine some me- thod by which, without essentially alter- ing the numerator, it may be made to shew the number of cyphers in the de- nominator. Something of the sort has already been done in representing a number of tens, hundreds, or thousands, &e.; for 5 thousands were represented by 5000, in which, by the assistance of cyphers, the 5 is made to stand in the place allotted to thousands. If, in the present instance, we place cyphers at the beginning of the numerator, until the number of figures and cyphers toge- ther is equal to the number of cyphers in the denominator, and place a point before the first cypher, the fraction bys numerator is 88, and whose denomina- tor is a decimal number containing four cyphers. There is a close connexion between the manner of representing decimal fractions, and the decimal notation for numbers. Take, for example, the frac- tion 217.3426, or 2173426 Voy will 10010 0 8 recollect that 2173426 is made up of 2000000 + 100000 + 70000 + 3000 + 400-+-20-+6. If each of these parts be divided by 10000, and the quotient obtained or the fraction reduced to its lowest terms, the result is as follows :— 4 2 6 zs roo + i000 10000° what it would have been had it come one place nearer to the decimal point. The first figure on the right hand of that point is so many tenths of a unit, the second figure so many hundredths of a unit, and so on. The learner should go through the same investigation with other fractions, and should demonstrate by means of the principles of fractions, generally, such exercises as the following, until he is thoroughly accustomed to this new me- thod of writing fractions :— _ 68342 = .6 + .08 + -003 + .0004 + .00002 68342 a 8 1 100 3 4 2 1000 7 T0000 + 100000 MATHEMATICS. 15 00012-= .0001 + .00002 = : : Bh. t ~ 10000 100000 2 163 29 16342 9 2. ies [ea cdedd slag er Miemace fp mag a NN ikea 1000 1000 +410 #£:11000 ~°»= 160 1000 The rules for addition, subtraction, and multiplication may now be under- stood. In addition and subtraction, the keeping the decimal points under one another is equivalent to reducing the fractions to a common denominator, as we may shew thus :—Take two fractions, 1.5 and 2.125, or 15 and 2135, which, reducing the first to the denominator of the second, may be written 159° and If we add the numerators toge- ther, we find the sum of the fractions $625, or 3.625 00? 2125 2.125 1500 1.5 3625 3.625 The learner can now see the connexion of the rule given for the addition of decimal fractions with that for the addi- tion of vulgar fractions. There is the same connexion between the rules of subtraction. The principle of the rule of multiplication is as follows :—If two decimal numbers be multiplied together, the product has as many cyphers as are in both together. Thus 100 x 1000 = 100000, 10 x 100 = 1000, &c. There- fore the denominator of the product, which is the product of the denomina- tors, has as many cyphers as are in the denominators of both fractions, and since the numerator of the product is the product of the numerators, the point must be placed in that product so as to cut off as many decimal places as are pen in the multiplier and multiplicand. hus 13 _—_—_—, 100 12. 156 10 10U0 “ 6 24 a I, Sap me, 1000 100 100000 or .004 x .06 = .00024, &e. It is a general rule, that wherever the number of figures falls short of what we know ought to be the number of deci- mals, the deficiency is made up by cyphers. It may now be asked, whether all fractions can be reduced to decimal fractions? It may be answered that they cannot. It is a principle which is demonstrated in the science of algebra, —that if a number be not divisible by a , or .13 x 1.2 =.156; prime number, no multiplication of that number, by itself, will make it so. Thus 10 not being divisible by 7, neither 10 x 10, nor 10 x 10 x 10, &ce. is divisible by 7. A consequence of this is, that since 5 and 2 are the only prime numbers which will divide 10, no fraction can be con- verted into a decimal unless its denomi- nator is made up of products, either of 5 or 2, or of both combined, such as SII DES OG 1 aye OS WM Op BRS Se) To shew that this is the case, take any fraction with such a denominator; for example, Multiply the nu- 13 OGp eran merator and denominator by 2, once for every 5, which is contained in the denomi- nator, and the fraction will then become 13 xX 2x2x2 2x2X2x13 5X 5X5X2K2X2 10X10 10" which is 404, or .104. In a similar way, any fraction whose denominator has no other factors than 2 or 5, can be reduced toa decimal fraction. We first search for such a number as will, when multiplied by the denominator, produce a decimal number, and then multiply both the numerator and denominator by that number. No fraction which has any other factor in its denominator can be reduced to a decimal fraction exactly. But here it must be observed, that in most parts of mathematical computation, a very small error 1s not material. In different spe- cies of calculations, more or less exact- ness may be required; but even in the most delicate operations, there is always a limit beyond which accuracy is useless, because it cannot be appreciated. For example, in measuring land for sale, an error of an inch in five hundred yards is not worth avoiding, since even if such an error were committed, it would not make a difference which would be con- sidered as of any consequence, as in all probability the expense of a more accu- rate measurement would be more than the small quantity of land thereby saved would be worth. But in the measure- ment of a line for the commencement of a trigonometrical survey, an error of an inch in five hundred yards would be fatal, because the subsequent processes involve calculations of such a nature that this error would be multiplied, and 16 cause a considerable error in the final result. Still, even in this case, it would be useless to endeavour to avoid an error of one-thousandth part of an inch in five hundred yards; first, because no instruments hitherto made would shew such an error: and secondly, because if they could, no material difference would be made in the result by a correction of it. Again, we know that. almost all bodies are lengthened in all directions by heat. For example :—A brass ruler which is a foot in length to-day, while it is cold, will be more than a foot to- morrow if it is warm. The difference, nevertheless, is scarcely, if at all, per- ceptible to the naked eye, and it would be absurd for a carpenter, in measuring a few feet of mahogany for a table, to attempt to take notice of it ; but in the measurement of the base of a survey, which is several miles in length, and takes many days to perform, it is neces- STUDY OF sary to take this variation into account, as a want of attention to it may produce perceptible errors in the result: never- theless, any error which has not this effect, it would be useless to avoid even were it possible. We see, therefore, that we may, instead of a fraction, which cannot be reduced to a decimal, substi- tute a decimal fraction, if we can find One so near to the former, that the error committed by the substitution will not materially affect the result. We will now proceed to shew how to find a series of decimal fractions, which approach nearer and nearer to a given fraction, and also that, in this approximation, we may approach as near as we please to the given fraction without ever being exactly able to reach it. Take, for example, the fraction =a. If we divide the series of numbers 70, 700, 7000, &c. by 11, we shall obtain the following results :— 70 gives the quotient 6, and the remainder 4, and is 6,4, moo pS 63 :, 7 63,1, ee as 636 a 4 6364. f Ei 0 * 6363 5 7 6363 1. &e. &e. Se. Now observe, that if two numbers do not differ by so much as 1, their tenth parts do not differ by so much as —1., their hundredth parts by so much as ,1., their thousandth parts by so much as =sl9p and so on; and also remember, that © is the tenth part of 4°, the hun- dredth part of 490, and so on. The two following tables will now be apparent :— <4 does not differ from 6 by so much as 1 700 i 63 ( 1 i : ; 0 53 636 3 1 Lonoe * 6363 phen &e &e. &e Therefore =; does not differ from ~6, or .6, by so much as qi, or .1 ms # Too -63 ” roo » Ol T41 2 Toon | 636 ” Toon » 001 an i io “ 6363. ,, rabvo ,» 0001 We have then a series of decimal frac- tions, viz.—.6, .63, .636, .6363, .63636, &c. which continually approach more and more near to =, and therefore in any calculation in which the fraction 7 appears, any one of these may be sub- stituted for it, which is sufficiently near to suit the purpose for which the calcu- lation is intended. For some purposes .636 would be a sufficient approximation; 7 for others .63636363 would be necessary. Nothing but practice can shew how far the approximation should, be carried in each case. The division of one decimal fraction by another is performed as follows :— Suppose it required to divide 6.42 by 1.213. The first of these is S42, and the second 1213, The quotient of these inary rule is 642000 oy 6420 by the ordinary rule is £42000, or 6420, MATHEMATICS. This fraction must now be reduced to a decimal on the principles of the last article, by the rule usually given, either exactly, or by approximation, according to the nature of the factors in the deno- minator. When the decimal fraction correspond- ing to a common fraction cannot be ex- actly found, it always happens that the se- ries of decimals which approximates to it, contains the same number repeated again and again. Thus, in the example which we chose, = is more and more nearly represented by the fractions .6, .63, .636, 6363, &c., and if we carried the process on without end, we should find a decimal fraction consisting entirely of repetitions of the figures 63 after the decimal point. Thus, in finding 1, the figures which are repeated ‘in the numerator are 142857. This is what is commonly called a cir- culating decimal, and rules are given in books of arithmetic for reducing them to common fractions. We would re- commend to the beginner to omit all notice of these fractions, as they are of no practical use, and cannot be tho- roughly understood without some know- ledge of algebra. It is sufficient for the student to know, that he can always either reduce a common fraction to a decimal, or find a decimal near enough to it for his purpose, though the calcu- lation in which he is engaged requires a degree of accuracy which the finest mi- croscope will not appreciate. But in using approximate decimals there is one remark of importance, the necessity for which occurs continually. Suppose that the fraction 2.143876 has been obtained, and that it is more than sufficiently accurate for the cal- culation in which it is to be employed. Suppose that for the object proposed it is enough that each quantity employed should be a decimal fraction of three places only, the quantity 2.143876 is made up of 2.143, as far as three places of decimals are concerned, which at first sight might appear to be what we ought to use, instead of 2.143876. But this is not the number which will in this case give the utmost accuracy which three places of decimals will admit of; the common usages of life will guide us in this case. Suppose a regiment consists of 876 men, we should express this in what we call round numbers, which in this case would be done by saying how many hundred men there are, leaving out of consideration the number 76, 17 which is not so great as 100; but in doing this we shall be nearer the truth if we say that the regiment consists of 900 men instead of 800, because 900 is nearer to 876 than 800. In the same manner, it will be nearer the truth to write 2.144 instead of 2.143, if we wish to express 2.143876 as nearly as possi- ble by three places of decimals, since it will be found by subtraction that the first of these is nearer to the third than the second. Had the fraction been 2.14326, it would have been best expressed in three places by 2.143; had it been 2.1435, it would have been equally well expressed either by 2.143 or 2.144, both being equally near the truth; but 2.14351 is alittle more nearly expressed by 2.144 than by 2.143. We have now gone through the lead- ing principles of arithmetical calculation, considered as a part of general Mathe- matics. With respect to the commercial rules, usually considered as the grand object of an arithmetical education, it is not within the scope of this treatise to enter upon their consideration. The mathematical student, if he is sufficiently well versed in their routine for the pur- poses of common life, may postpone their consideration until he shall have become familiar with algebraical opera- tions, when he will find no difficulty in- understanding the principles or practice of any of them. He should, before com- mencing the study of algebra, carefully review what he has learnt in arithmetic, particularly the reasonings which he has met with, and the use of the signs which have been introduced. Algebra is at first only arithmetic under another name, and with more general symbols, nor will any reasoning be presented to the student which he has not already met with in establishing the rules of arithmetic. Huis progressin the former science depends most materially, if not altogether, upon the manner in which he has attended to the latter; on which account the detail into which we have entered on some things which to an in- telligent person are almost self-evident, must not be deemed superfluous. When the student is well acquainted with the principles and practice of arith- metic, and not before, he should com- mence the study of algebra. It is usual to begin algebra and geometry together, and if the student has sufficient time, if is the best plan which he can adopt. Indeed, we. see no reason why the elements of geometry should ae precede 18 those of algebra, and be studied toge- ther with arithmetic. In this case the student should read that part of these treatises which relates to geometry, first, It is hardly necessary to say that though we have adopted one particular order, yet the student may reverse or alter that order so as to suit the ar- rangement of his own studies. We now proceed to the first prin- ciples of algebra, and the elucidation of the difficulties which are found from experience to be most perplexing to the beginner. We suppose him to be well acquainted with what has been pre- viously laid down in this treatise, par- ticularly with the meaning of the signs +, —, xX, and the sign of division. Cuapter VI. Algebraical Notation and Principles. WHENEVER any idea is constantly re- curring, the best thing which can be done for the perfection of language, and consequent advancement of knowledge, is to shorten as much as possible the sign which is used to stand for that idea. All that we have accomplished hitherto has been owing to the short and ex- pressive language which we have used to represent numbers, and the opera- tions which are performed upon them. The first step was to write simple signs for the first numbers, instead of words at full length, such as 8 and 7, instead of eight and seven. The next was to give these signs an additional meaning, according to the manner in which they were connected with one another; thus 187 was made to represent one hundred added to eight tens added to seven. The STUDY OF | next was to give by new signs directions when to perform the operations of ad- dition, subtraction, multiplication, and division ; thus 5+8 was made to re- present 8 added to 5, and soon. With these signs reasonings were made, and truths discovered which are common to all numbers; not at once for every number, but by taking some example, by reasoning upon it, and by producing a result; this result led to a rule which was declared to be a rule which held equally good for all numbers, because the reasoning which produced it might have been applied to any other example as well as to the one which was chosen. In this way we produced some results, and might have produced many more ; the following is an instance :—half the sum of two numbers added to half their difference, gives the greater of the two numbers. For example, take 16 and 10, half their sum is 13, half their dif- ference is 3; if we add 13 and 3 we get 16, the greater of the two numbers. We might satisfy ourselves of the truth of this same proposition for any other num- bers, such as 27 and 8, 15 and 19, and soon. If we then make use of signs, we find the following truths :— 16+10 , 16—10 sport heserenggane | $e 27+8 27-8 = 27. 2 i 2 / was +25 = 15, and so on. If, then, we choose any two numbers, and call them the first and second num- bers, and call that the first number which is the greater of the two, we have the following :— First N° — second N°? First N°-+ second N° as + In this way we might express anything which is true of all numbers, by writing first N°, second N°, &c., for the dif- ferent numbers which enter into our proposition, and we might afterwards _ TF fe) 9 ao Hivet IN suppose the first N°, the second N°, &ce. to be any which we please. In this way we might write down the following assertion, which we should find to be always true :— (First N°-+ second N°) multiplied by (First N°— second N°) = First N° x first N° — Second N° x second N°. When any sentence expresses that two numbers or collections of numbers are equal to one another, it is called an equation, thus 7+5=12 is an equation, and the sentences which have been just written down are equations. Now the next question is, could we not avoid the trouble of writing first N°, second N°, &e. so frequently ? This is done by putting letters of the alphabet to stand for these numbers. Suppose, for example, we let x stand for the first number, and y for the second, the two assertions already made will then be written thus :-— MATHEMATICS. ety, ©-y PT he (w+y) X (w—y) = UXX—YXY. By the use of letters we are thus enabled to write sentences which say something of all numbers, with a very small part only of. the time and trouble necessary for writing the same thing at full length. We now proceed to enume- rate the various symbols which are used. 1. The letters of the alphabet,are used 19 to stand for numbers, and whenever a letter is used it means either that any number may be used instead of that letter, or that the number which the letter stands for is not known, and that the letter supplies its place in all the reasonings until it is known. 2. The sign + is used for addition, as in arithmetic. Thus 7+ is the sum of the numbers represented by x and 2. The following equations are sufficiently evident :— LUtYrZ = a“+tety =yte2t+a. Ifa=6,thenat+e=b+c,atct+d=b+c4d, &e. 3. The sign — is used for subtraction, tions will show its use :— as in arithmetic. The following equa- ttra-b-cte=r+tatve=—b—eH=aete—b+-x. Ifa=ba—c=b—c,a—c+d=b—c+d, &e. 4, The sign x is used for multiplica- tion as in arithmetic, but when two numbers represented by letters are mul- tiplied together it is useless, since a x b can be represented by putting a and 6 together thus, a 6. Also ax6xXc is re- presented by abc;\a xax a, for the present we represent thus, aaa. When two numbers are multiplied together, it is necessary to keep the sign x ; other- wise 7 x5 or 35 would be mistaken for 75. It is, however, usual to place a ‘point between two numbers which are to be multiplied together; thus 7x5 x3 is written 7.5.3. But this point may ‘sometimes be mistaken for the decimal ‘point: this will, however, be avoided by always writing the decimal point at the head of the figure, viz. by writing 23461 thus, 234°61. 5. Division is written, as in arithmetic ; thus, : signifies that the number repre- sented by a is to be divided by the num- ber represented by 0. 6. All collections of numbers are ‘called expressions ; thus, a+ 6,a+b—c, aa+bb—d, are algebraical expres- sions. [exe pecs: X00 Bea en CBB eh ee I, LXBXL KD DLUL , a, LeXLXUXLXL wexwe, . . Ww, . : Se. &e. Se. There is no point which is so likely to create confusion in the ideas of a begin- mer as the likeness between such ex- pressions as 4a and a*., On this ac- count.it would be better for him to omit 7. When two expressions are to be multiplied together, it is indicated by placing them side by side, and inclosing each of them in brackets. Thus, if a+b+c is to be multiplied by d+e+/f/, the product is written in any of the fol- lowing ways :— (a+b+c)(d+et+/), fatb+cl[d+et+/], atb+c.,dtetf, at+b+c).d+e+/f. 8. That a is greater than & is written thus, a > 6. 9, That a is less than 6 is written thus, a < 0. 10. When there is a product in which all the factors are the same, such as avxxvxre,which means that five equal num- bers, each of which is represented by 2, are multiplied together, the letter is only written once, and above it is written the number of times which it occurs, thus, xvxxx is written x. The following table should be carefully studied by the student :— Ss or xx is written «2, and is called the square, or second power of a: . cube, or third power of x. . . . fourth power of x. . fifth power of x._ Se. using the latter expression, and to put xxaxux in its place until he has acquired some little facility in the operations of algebra. If he does not pursue this course he must recollect soa 7% 4, m 20 STUDY OF these two expressions, has different The difference of meaning will be appa- names and meanings. In 4a itiscalled rent from the following tables :— a coefficient, in x* an exponent or index, 27 = 24-0 8@=xa7+ane+nr © Av=eto+orer &e. C= ox Parr, Lark exe or v7; Cx OXBXL XL OY POLTT, &e. TiweS Mrs 6) w= 9: ACEO Tae =O7, 4v=12 w=8r, &e. &e. The beginner should frequently write for himself ‘such expressions as the following :— 4a’ b®=aaabb + aaahb + aaabb + aaabb 5a* x=aaaax + aaaax + aaaaxv+ aaaax+ aaaax 9a? 6? +-4ab* = 9aabbb+4abbbb GB aa+bb aa — Pg bb cae Dp bb | aa- te bb+ ec aa-6bb aa—bb aa—bd eO-b aaa— bbb _ aa + ab+bb bt OTe bb With many such expressions every book on algebra will furnish him, and he should then satisfy himself of their truth by putting some numbers at pleasure instead of the letters, and making the results agree with one another. Thus, to 3 23 try the expression : : = @ +ab+6%, or, which is the same, ae ns aa-+-ab+6b. Leta stand for 6 and } stand for 4, then, if this expression be 6.6.6 —4.4.4 § ea Ga 6.6+6.4+4.4, which is correct, since each of these expres- Slons is found, by calculation, to be 76. The student should then exercise himself in the solution of such ques- tions as the following : — What is ati Baus ab a a+b ', ab ins a stands for 6, and 6 for 5, JI. when a stands for 13, and 6 for 2, and so on. He should stop here until he has, by these means, made the signs familiar to his eye and their meaning to his mind: nor should he proceed to any further algebraical operations until he can rea- dily find the value of any algebraical ex- pression, when he knows the numbers which the letters stand for. He cannot, at this period of his course, write too many algebraical expressions, and he must particularly avoid slurring over the sense of what he has before him, and must write and rewrite each expression true, a, I. when a+b until the meaning of the several parts forces itself upon his memory at first sight, without even the necessity of putting it in words. It is the neglecting to do this which renders the operations of algebra so tedious to the beginner. He usually proceeds to the addition, sub- traction, &c. of symbols, of the meaning of which he has but an imperfect idea, and which have been newly introduced to him in such numbers that perpetual confusion is the consequence. We can- not, therefore, use too many arguments to induce him not to mind the drudgery of reducing algebraical expressions into figures. This is the connecting link be- tween the new science and arithmetic, and, unless that link be well fastened, the knowledge which he has previously acquired in arithmetic will help him but little in acquiring algebra. | The order of the terms of any alge- braical expression may be changed with- out changing the value of that expres-— sion. This needs no proof, and the fol- lowing are examples of the change :— _ a+b+ab+c+ac—d—e—de—f =a—d+b—e+ab—de+c—f+ac . =a+b—d—e—de—f+ac+c+ab : =ab+ac—de+a+6+c—e-f-d When the first term changes its place, as in the fourth of these expressions, the — sign++is put before it, and should, pro-_ perly speaking, be written wherever there is no sign, to indicate that the — term in question increases the result. of F the rest, but it is usually omitted. The . MATHEMATICS. negative sign is often written before the first term, as in the expression—a+6: but it must be recollected that this is written on the supposition that a@ is sub- tracted from what comes after it. When an expression is written in brackets,’ with some sign before it, such as a—(b—c), it is understood that the expression in brackets is to be consi- dered as one quantity, and that its re- sult or total is to be connected with the a—(b-—c)=a-b+e Similarly 21 rest by the sign which precedes the brackets. In this example it is the difer- ence of b ande which is to be subtracted froma. If a=12, b=6, and c=4, this is 10. In the expression a—b made by subtracting 5 from a, too much has been subtracted by the quantity c, since it is not 5b, but 6—c, which must be sub- tracted from a. In order, therefore, to make a — (6 —c) c¢ must be added to a~6, which givesa—b+c, Therefore, a a+b—(c+d—e—f)=atb—c—dt+e+f (az?’—bx +c)—(da?—ext+f)=ax*~—bx+c—dx’?+ex—f. When the positive sign is written before an expression in brackets, the brackets may be omitted altogether, unless they serve to show that the expression in question is multiplied by some other. Thus, instead of (a+b+c)+(d+et+f), we may write a+b+c+d+e+ f/f, which is, in fact, only saying that two wholes may be added together by adding together all the parts of which they are composed. But the expression a+(d+c) (d+e) must not be written thus: a+ b+c(d+e), since the first expresses that (6+c) must be multiplied by (d+ e) and the product added to a, and the second that c must be multiplied by (d+ e) and the product added toa+6. Ifa, b, c,d, ande, stand for 1, 2, 3, 4, and 5, the first is 46 and the second 30. When two. or more quantities have exactly the same letters repeated the same number of times, such as 4a? 6°, and 6a” 6°, they may be reduced into one by merely adding the coefficients and re- taining the same letters. Thus, 2a+3a is 5a, 6bc—4bc is 2be, 3 (w+ y)+2(v+y) is5 (a+y). These things are evident, but beginners are very liable to carry this farther than they ought, and to at- tempt to reduce expressions which do not admit of reduction. For example, they will say that 3b+46° is 46 or 40°, neither of which is true, except when 6 stands for 1. The expression 36+ 62, or 36+68, cannot be made more simple until we know what 6 stands for. The following table will, perhaps, be of service. 6 a 6+ 3 a bis not 9 a’ b° 6a—4@ is not 2 a 26a+ 3b is not 5 ab Such are the mistakes which beginners almost universally make, mostly for want of a moment’s consideration. They attempt to reduce quantities which can- not be reduced, which they do by adding the exponents of letters as well as their coefficients, or by collecting several terms into one, and leaving out the signs of addition and subtraction. The be- ginner cannot too often repeat to him- self that two ferms can never be made into one, unless both have the same let- ters, each letter being repeated the same number of times in both, that is, having the same index in both. When this is the case, the expressions may be reduced by adding or subtracting the coefficients according to the sign, and affixing the common letters with their indices. When there is no coefficient, as in the expres- sion a 6, the quantity represented by a® 6 being only taken once, 1 is called the coefficient. Thus, 3ab+4ab+6ab —ab—7av0=5a0 Oy Rey Oy ey = Oy The student must also recollect that he is not at liberty to change an index from one letter to another, as by so doing he changes the quantity represented. Thus a*b and ad* are quantities totally distinct, the first representing aaaad and the se- cond abbbb. The difference in all the cases which we have mentioned will be made more clear, by placing numbers at pleasure instead of letters in the expres- sions, and calculating their values ; but, in conclusion, the following remark must be attended to. If it were asserted that a” + 5° a+t+6 , and we wish to pro- the expression is the same as 15 2 ab iY be 2a—0 ceed to see whether this is always the case or no, if we commence accidentally by supposing 6 to stand for 2 and a for 4, we shall find that the first is the same as the second, each being 33. But we must not conclude from this that they are always the same, at least until we 22 have tried whether they are so, when other numbers are substituted for a and 6. If we place 6 and 8 instead of a and 6, we shall find that the two ex- pressions are not equal, and therefore we must conclude that they are not always the same. Thus in the expres- sions 32 —4 and 2x2 + 8, if x stand for 12, these are the same, but if it stands for any other number they are. not the same, Cuapter VII. Elementary Rules of Algebra. Tue student should be very well ac- quainted with the principles and notation hitherto laid down before he proceeds to the algebraical rules for addition and subtraction. He should then take some simple examples of each, and proceed to find the sum and difference by reason- ing as follows. Suppose it required to add e—d to a— 6, The direction to do this may either be written in the com- mon way thus: a —6 c—d Add or more properly thus: Find (@ — 4)+ (c—d). If we add c to a, or find a+c, we have too much ; first, because it is not a which is to be increased by c — d but a — b; this quantity must therefore be decreased STUDY OF by 6 on this account, or must become a+c—6; but this is still too great, because it is not c which was to be added but ec —d; it must therefore be decreased by d on this account, or must become a + c—b—dora—b+e—d. From a few reasonings of this sort the rule may be deduced; and not till then should an example be chosen so compli- cated.as to make the student lose sight for one moment of his demonstration, The process of subtraction we have already performed, and from a few ex- amples of this method the rule may be deduced. The multiplication of a by c—d is performed thus: ais to be taken c —d times. Take it first c times or find ac, This is too great, because a has been taken too many times by d. From ac we must therefore subtract d times a, or ad, and the result is that a(e— dad) =ac — ad. This may be verified from arithmetic, in which the same process is shewn to be correct ; and this whether the numbers a,c, and d are whole or fractional. For example, it will befound that 6 (14 — 9) or 6 X 5 isthesameas 6 x 14—6 x 9, a(b+c—d)= ab + ac —ad. (p + pq — ar) rz = pxz — pqxrz — arxz. (a2 + 25%) 6%, or (aa + 266) bb = aabb + 2bdbb. =a b?-+- 26+, Also when a multiplication has been performed, the process may be reversed and the factors of it may be given. Thus, on observing the expression ab — ac+a?, or ab—ac-+aa, we see that in its for- mation every term has been multiplied by a; that is, it has been made by mul- tiplying 6 —c+ a by a, or aby b—c ~~ 4. There will now be no difficulty in perceiving that ac + ad + bc + bd=a(e+da)+b(e+d)= (a+ b)(c+d) a® — ab? + 2abe —de + 3a = a(a — 62 + 3) +¢(2ab —d) It is proved in arithmetic that if num- bers, whether whole or fractional, are multiplied together, the product remains the same when the order in which they are multiplied is changed. Thus 6x4x3=3 x 6x 4=4 xX 6 x 8, &e,, and uk 4 eo ip, We &c. Also, thata part of the multiplication may be made, and the partial product substituted ins stead of the factors which produced it, thus, 3x 4x5 x6 is 12x5x6, or 15 x 4 x 6, or 90 x 4.° From these rules two complicated single terms may be multiplied together, and the product represented in the most simple manner. which the case admits of. Thus if it be required to multiply 6 a’ 6+ c, which is 6aaabbbbc by 12a%d%c%d, which is 12aabbbcecd, the product is written thus: 6aaabhbbc 12aabbbeced, which multiplication may be performed in the following order: 6 X 12aaaaabbbbbbbccccd, which is re- presented by 72a°b’c'd. A few exam- ples of this sort will establish the rule for the mulfiplication of such quantities which is usually given in the treatises on Algebra. It is to be recollected that in every algebraical formula which is true of all numbers, any algebraical expression MATHEMATICS, may be substituted for one of the letters, provided care is taken to make the sub- stitution wherever that letter occurs thus from the formula: a2? — b? = (a + b) (a — BD) we may deduce the following by making 23° substitutions for a. If this formula be always true, it is true when a is equal top+q, that is, it istrue if p+ q be put instead of a@ wherever that letter occurs in the formula. Therefore, p+q?—&=(p+q+d) ~+q-b). ‘Similarly, 6 + m)\? — 0% = (2b-+- m)m aty\-e%-y)\?=2+y Te -y| ery =(@ = )\ . = 4xy, and so on. ~ We have already established the formula, (p — 4) a = ap — aq. Instead of a let us put 7 — s, and this formula becomes. (p-Qr—s)=r-slp—r-s\q. But 7 — s\p = pr — ps, and r — slg = qr = qs. Therefore p = q) 7 — s! = pr —ps — (qr — qs) = pr — ps = qr + Qs By reasoning in the same way we may prove that | p-q\r--s\= pr--ps — qr = 9. A few examples of this sort will esta- blish what is called the rule of signs in multiplication; viz. that a term of the multiplicand multiplied by a term of the multiplier has the sign-+ before it ifthe terms have the same sign, and — if they have different signs. But here the stu- dent must avoid using an incorrect mode of expression, which is very common, viz; the saying that + multiplied by + gives +; — multiplied by + gives —; andsoon. He must recollect that the signs -+ and — are not quantities, but directions to add and subtract, and that, as has been well said by one of the most luminous writers on algebra im our language, we might as well say, that take away multiplied by take away gives add, as that — multiplied by - gives--*. | The only way in which the student should accustom himself to state this rule is the following: “In multiplying two algebraical expressions, multiply each term of the one by each term of the other, and wherever two terms are pre- ceded by the same sign put ++ before the product of the two; when the signs are different put the sign — before their product.” If the student should meet with an * Frend, Principles of Algebra, The author | of this treatise is far from agreeing with the work which he has quoted in the rejection of the isolated negative sign which prevails throughout it, but fully concurs in what is there said of the methods then in use for explaining the difficulties of the nega- five sign. ; equation in which positive and negative signs stand by themselves, such as +ab x —c= — abe, let him, for the present, reject the exam- ple in which it occurs, and defer the consideration of such equations until he has read the explanation of them to which we shall soon come. Above all, he must reject the definition still some- times given of the quantity — a, that it is less than nothing. It is astonishing that the human intellect should ever have tolerated such an absurdity as the idea of a quantity less than nothing; above all, that the notion should have out- lived the belief in judicial astrology and the existence of witches, either of which is ten thousand times more possible. These remarks do not apply to such an expression as — 6 + a; which we sometimes write instead of a — 6, as long as it is recollected that the one is merely used to stand for the other, and for the present a must be considered as greater than 6. In writing algebraical expressions, we have seen that various arrangements may be adopted. Thus aa? — br+e may be written as c + aa — ba, or — bz +e +ax*. Of these three the first is generally chosen, because the highest power of x is written first; the highest but one comes next ; and last of all the term which contains no power of a. When written in this way the ex- pression is said to be arranged in 24 descending powers of 2; had it been written thus, c —bx% + ax*%, it would have been arranged in ascending powers of x2; in either case it is said to be arranged in powers of w, which is called the principal letter. It is usual to arrange all expressions which occur in the same question in powers of the same letter, and practice must dictate the most con- venient arrangement. Time and trouble is saved by this operation, as will be evident from multiplying two unarranged expressions together, and afterwards doing the same with the same expres- sions properly arranged. STUDY OF In multiplying two arranged expres- sions together, while collecting such terms into one as will admit of it, it will always be evident that the first and last of all the products contain powers of the principal letter which are found in no other part, and stand in the product unaltered by combination with any other terms, while in the interme- diate products there are often two or more which contain the same power of the principal letter, and can be reduced into one. This will be evident in the following examples: Multiply . . . 26 —3 4+ x ByisP rs vi—-2a%°9+an The product is . w0~— 3 49 + 28 Or ° e s & ® Multiply . . DY ii nthe The product is ax? + bar + cx dx? + er +f —2 72+ 6 #7 —2 76 +a7 —3 x6 + x5 x03 79 ~784+727 — 5 #6 + 75, adx5 + bdax*t + cdx8 + aext + hex? + cex2 afxe + bfx® + cfx Or . . « . adae+(bd+ae)a*+(cd+betaf) 2+ (ce+ bf) a2+efx. It is plain from the rule of multiplica- tion, that the highest power of x in a product must be formed by multiplying the highest power in one factor by the ~ highest power in the other, or when the two factors have been arranged in de- scending powers, the first power in one by the first power in the other. Also, that the lowest power of 2, or should it so happen, the term in which there is no power of x, is made by multiplying the last terms in each factor. These being the highest and lowest, there can be no other such power, consequently neither of these terms can coalesce with any other, as is the case in the intermediate products. This remark will be of most convenient application in division, to which we now come. Division is in all respects the reverse of multiplication. In dividing a by b we find the answer to this question ;—if a be divided into 6 equal parts, what is the magnitude of each of those parts ? The quotient is, from the definition of a fraction, the same as the fraction and all that remains is to see whether that fraction can be represented by a simple algebraical expression without fractions or not; just as in arithmetic the division of 200 by 26 is the reduction of the frac- tion %°.9 to a whole number, if possible. But we must here observe that a dis- tinction must be drawn between alge- braical and arithmetical fractions. For at+é., 4 , example, a 1s an algebraical fraction, that is, there is no expression without fractions which is always equal to pa a ee Ne But it does not follow from this that the number which aoe represents, is always an arithmetical fraction; the contrary may be shown. Let a stand for 12, and 6 for 6, then — Se Ke Again, a@+ab is a quantity which does not contain algebraical fractions, but it by no means follows that it may not re- present an arithmetical fraction. To show that it may, let a=1 and b=2, then a+ab=14 or 5, Other examples will clear up this point if any doubt yet exist in the mind of the student. Never- theless, the following propositions of arithmetic and algebra, which only differ in this, that ** whole number” in the arithmetical proposition is replaced by MATHEMATICS. * simple expression” * in the algebraical one, connect the two subjects and render those demonstrations which are in arith- The sum, difference, or product of two whole numbers, is a whole number. One number is said to be a measure of another when the quotient of the two is a whole number. The greatest common measure of two whole numbers is the greatest whole number which measures both, and is the product of all the prime numbers which will measure both, When one number measures two others, if measures their sum, difference, and product. ' In the division of one number by another, the remainder is measured by any number which measures the dividend and divisor. A fraction is not altered by multiply- ing or dividing both its numerator and denominator by the same quantity. In the term simple expression are in- cluded those quantities which contain arithmetical fractions, provided there is no algebraical quantity, or quantity re- presented by letters in the denominator ; thus 4 ab+4 is called a simple expres- sion. We now proceed to the division of one simple expression by another, and we will take first the case where neither quantity contains more than one term. For example, what is 42 a*bc divided by 6 a? 6c? that is, what quan- tity must be multiplied by 6 a? dc, in order to produce 42 a* 0c, This last expression written at length, is 42 aaaa bbb c, and 42 is 6x7. We can then separate this expression into the ‘product of two others, one of which shall be 6 a? bc, or 6 aa bc; it will then be 6 aa bc x 7 aa bb, and it is 7 aabb which must be multiplied by 6 aa dc in ‘order to produce 42 a* 03 c. A few ex- amples worked in this way, willlead the student to the rule usually given in all ‘cases but one, to which we now come. ‘We have represented cc, ccc, cecc, &e. by c?, c, ct, &c., and have called them the second, third, fourth, &c. powers of c. The extension of this rule would lead us to represent ¢ by c', and call it the first power ofc. Again, we have represented e+c, c+c+c, ctc+cte, +&c. by 2c, 3c, 4c, and have called * By a simple expression is meant one which does not contain the principal letter in the denominator of any fraction. 25 metic confined to whole numbers, equally true in algebra as far as regards simple expressions :— The sum, difference, or product of two simple expressions is a simple expression. One expression is said to be a measure of another when the quotient of the two is a simple expression. The greatest common measure of two expressions is the common measure which has the highest exponents and coefficients, and is the product of all ee simple expressions which measure oth. When one expression measures two others, it measures their sum, difference, and product. In the division of one expression by another, the remainder is measured by any expression which measures the dividend and divisor. A fractional expression is not altered by multiplying or dividing both its nu- merator and denominator by the same expression. 2, 3,4, &e. the coefficients of c. The extension of this rule would lead us to write c thus, Ic, or, rather, if we at- tend to the last remark, 1c. This in- stance leads us to observe the gradual progress of our language. We begin with the quantity c by itself; we pro- ceed in our course, shortening by new signs the more complicated combi- nations of c, and the original quantity c forces itself anew upon our atten- tion as a part of the series, Cy 2c, 3c, 4c, &e., and c, c?, 8, ct, &e. in each of which, except the first, there is a distinct figure, which is called a coefficient or exponent, according to its situation. -We then deduce rules in which the terms coefficient or exponent occur, but which, of course, cannot apply to the first term in each series, because, as yet, it has neither coefficient nor exponent. Among such rules are the following :— To add two terms of the first series, add the coefficients, and affix to the sum the letter c. Thus 4¢+3c=7c. To multiply two terms of the second series, add the exponents, and make this sum the exponent ofc. Thus cx ce=c!. To divide a term of the second series by one which comes before it, subtract the exponent of the divisor from the ex- ponent of the dividend, and make this difference the exponent ef c. Thus, 26 These rules are intelligible for all terms of the series except the first, to which, nevertheless, they will apply if we agree that 1c! shall represent ¢, as will be evident by applying either of, : 4 : them to find 4c+c, c*xe, or =. We therefore agree that 1c! shall stand for ¢, and although ¢ is not written thus, it must be remembered that c is to be con- sidered as having the coefficient 1 and the exponent 1, which is an amendment and enlargement of our algebraical lan- guage, derived from experience. It may be said that this is all superfluous, be- eause, if c2 stand for ec, and c? for cce, what can c! stand for but ¢? But it must be recollected that, smce the sym- bol & has not yet received a meaning, we are at liberty to make it stand for anything which we please, for example, If we did this, there would, indeed, be a great violation of analogy, that is, what e} stands for would not be as like that which c? has been made to stand for, as the meaning of ¢ is to that of c*; but, nevertheless, we should not be led to any incorrect results as long as we re- membered to make c' always stand for the same thing. These remarks are here introduced in order to show the manner in which analogy is followed in extending the language of algebra, and to prove that, after a certain period, we may rather be said to discover new sym- bols than to make them. The immense importance of this branch of the subject makes it necessary that it should be fully and early understood by all who intend to pursue their mathematical stu- dies to any depth. To illustrate it still further, we subjoin another instance, which has not been noticed in its proper place. The signs + and—were first used to connect one quantity with others, and to show what arithmetical operations were performed on other quantities by means of the first. But the first quan- tity on which we begin the operation is not preceded by any sigh, not being considered as added or subtracted to any previous one. Rules were afterwards deduced for the addition and subtraction of the total result of several expressions in which these signs occur, as follows: To add two expressions, form a third, which has all the quantities in the first two, with the same signs, 1+6é for Z , or c—c%, or any other. STUDY OF To subtract one expression from ano-' ther, change the sign of each term of the subtrahend, and proceed as in the last rule. The only terms in which these rules, do not apply are those which have no sign, viz: the first of each. But they, will apply to those terms, and will pro-) duce correct results, if we place the sign + before each of them. We are. thus led to see that an algebraical term which has no sign is equivalent in all operations to one which is preceded by) the sign +. We, therefore, consider, this sign as prefixed, though it is not always written, and thus we are fur-, nished with a method of containing” under one rule that which would other-— Wise require two. ¥ From these considerations the follow-_ ing appears to be the best and most na-_ tural course of proceeding: in the inven-— tion of additional symbols. When a rule has been discovered which is not” quite general, and which only fails in” its application to a few instances, annex — such additional symbols to those already © in use, or change and modify these so as’ to make the rule applicable in all cases, provided always this can be done with-— out making the same symbol stand for two different things, and without any violation of analogy. If the rule itself, by its application to any case, should produce anew symbol hitherto unex- — plained, it is a sign that the rule has been applied to a case which was never intended to fall under it when it was_ made. For the solution of this case we must have recourse to first principles, © but when, by these means, the result has been found, it will be best to agree that the new symbol furnished by the rule shall stand for the result furnished © by the principle, by which mieans the generality of the rule will be attained and the analogy of language will not be © injured. Of this the following is a res markable instance :— To divide c® by c5 the rule tells us to subtract 5 from 8, and make the result the exponent of c, which gives the quo- tient c%. If we apply the same rule to divide c6 by ¢®, since 6 subtracted from 6 leaves 0, the result is c°, a new symbol, to which we have attached no meaning, © The fact is that the rule was formed © from observation of different powers of ¢, and was never intended to apply to the di- | vision of a power of ¢ by the same power. — If we apply the common principles to the division of c® by cS, the result is 1. We, | | | : MATHEMATICS. therefore, agree that c’ shall stand for 1, 27 quotient found by the divisor,"and sub- and the least inspection will show that this agreement does not affect the truth of any result derived from the rule. If, in the solution of any problem, the symbol c° should appear, we must con- sider itisa sign that we have, in the course of the investigation, divided a power of c by itself by the common rule, without remarking that the quotient is 1. We must, therefore, replace c’ by 1, but it is entirely indifferent at what stage of the process this is done. ate Several extensions might be noticed, which are made almost intuitively, to which these observations will apply. Such, for example, is the multiplication and division of any number by 1, which” is not contemplated in the definition of these operations. Such is also the con- tinual use of 0 as a quantity, the addition and subtraction of it from other quanti- ties, and the multiplication of it by others, neither of which were contem- plated when these operations were first thought of. We now proceed to the principles on which more complicated divisions are per- formed. The question proposed in divi- sion, and the manner of answering it, may be explained in the following manner. Let A be an expression which is to be divided by H, and let Q be the quotient of the two. By the meaning of division, if there be no remainder A = QH, since the quotient is the expression which must multiply the divisor, in order to produce the dividend. Now let the quo- tient be made up of different terms, a, b, c, &e., let it be at b—et+d. That is, let A=QH (1) Q =at b eC +d (2) By putting, instead of Q in (1), that which is equal to it in (2), we find A=(@atb—-c+aaH = aH+5H—cH+dH (3) Now suppose that we can by any method find the term a of the quotient, that is, that we can by trial or otherwise find one term of the quotient. In (3), when the term a is found, since H is known, the term a@H is found. Now if two quantities are equal, and from them we subtract the same quantity, the remain- ders will be equal. Subtract aH from the equal quantities A and aH + bH— cH + dH, and we shall find A—aH=0H-cH+dH= (6—ce+d) H. (4) If, then, we multiply the term of the tract the product from the dividend, and call the remainder B; then - B =(6—c+d)H. (5) That is, if B be made a dividend, and H still continue the divisor, the quotient is 6 —c-+d, or all the first quotient, ex- cept the part of it which we have found. We then proceed in the same manner with this new dividend, that is, we find 6 and also 6H, and subtract it from B, and let B —OH be represented by C, which’ gives by the process which has just been explained C=(—ce+aH=-—-cH+dH. (6) We now come to a negative term of the quotient. Let us suppose that we have found c, and that its sign in the quotient is —. If two quantities are equal, and we add the same quantity to both, the sums are equal. Let us there- fore add cH to both the equal quantities in (6), and the equation will become C+cH =dH; (7) or if we denote C + cH by D, this is D=dH. There is only one term of the quotient remaining, and if that can be found the process is finished. But as we cannot know when we have cometo the last term, we must continue the same process, that is, subtract dH from D, in doing which we shall find that dH is equal to D, or that the remainder is nothing. This indicates that the quotient is now ex- hausted and that the process is finished. We will now apply this to an exam- ple in which the quotient is of the same form as that in the last process, namely, consisting of four terms, the third of which has the negative sign. Thisis the division of. Y= 3 y+ ByteZrybyx—y. Arrange the first quantity in descending powers of w which will make it stand thus— B+ eB8y—32°y+2ryr—y* (A) One term of the quotient can be found immediately, for since it has been shewn that the term containmg the highest power of x in a product, is: made up of nothing but the product of the terms containing the highest powers of # which occur in the multiplier and multiplicand, and considering that the expression (A) is the product of x — y and the quotient, we shall recover the highest power of x in the quotient by dividing x*, the highest power of # im (A), by @, its highest 28 power ina —y. This division gives 2’ as the first term of the quotient. The following is the common process, and (H) STUDY OF with each line is put the corresponding — step of the process above explained, of which this is an example :-— (A) (a) L—-y V+ Vy—3L y+ Ary’ — y* («° (aH) Subtract a‘ xy | | (6) (B) Second dividend . 222y —3aty® + 2ry—y* (4+ 2ary (6H) Subtract . 2aey—2a*y® Ps c (C) Third dividend . 22 2 — ey? + 22y— y' (—ay? (cH) SUDLTACE poy 4 all eae aie meat (d) (D) Fourth dividend . . Ni A ry — yt (+ ¥ (d H) Subtnact... .. i Abs eth Ura 0 The whole quotient is therefore 2° + 2 a2 y — xy? + y°. The second and following terms of the quotient are determined in exactly the same manner as the first. In fact this process is not the finding of a quotient directly from the divisor and dividend, but one term is first found, and by means of that term another dividend is obtained, which only differs from ihe first in having one term less in the quotient, viz. that which was first found. From this second dividend one term of its quotient is found, and so on until we obtain a dividend whose quotient has only one term, the finding of which finishes the process. It is usual also to neglect all the terms of the first dividend, except those which are immediately wanted, taking down the others one by one as they become necessary. This is a ver good method in practice, but should be avoided in explaining the principle, since the first subtraction is made from the whole dividend, though the operation may only affect the form of some part of it. If the student will now read atten- tively what has been said on the greatest common measure of two numbers, and then examine the connexion of whole numbers in arithmetic and simple ex- pressions in algebra, with which we commenced the subject of division, he will see that the greatest algebraical common measure of two expressions may be found in exactly the same man- ner as the same operation is performed in arithmetic. He must also recollect that the greatest common measure of two expressions A and B is not altered by multiplying or dividing either of them, A for example, by any quantity, provided that quantity has no measure in com- mon with B, For example, the greatest common measure of a®—a? and ba'— bx’? is the same with that of 2a®—22° and a®’— 2’, since though a new measure is now introduced into the first, and taken away from the second, nothing is introduced or taken away which is common to both. The same observation applies to arithmetic also. For example, take the numbers 162 and 180. We may, without altering their greatest common measure, multiply the first by 7 and the second by 11, &c. The rule for finding the greatest com- mon measure should be practised with great attention by all who intend to pro- ceed beyond the usual stage in algebra. To others it is not of the same import- ance, as the necessity for it never occurs in the lower branches of the science. In proceeding to the subject of frac- tions, it must be observed that, in the same manner as in arithmetic, when there is a remainder which cannot be further divided by the divisor, that is, where the dividend is so reduced that no simple term multiplied by the first term of the divisor will give the first term of the remainder, as in the case where the divisor is a?a + ba and the remainder ax + 6; in this case a frac- tion must be added to the quotient, whose numerator is this remainder, and whose denominator is the divisor. Thus in dividing a* + 5* by a+ 6, the quotient is a —a’ 6 +ab*?+ 6%, and the remainder 26+, whence ae oles was 254 ee Pe ae oe et ae MATHEMATICS. The arithmetical rules for the addition, &c. of fractions hold equally good when the numerators and denominators are themselves fractions. Thus # and? are 7 2 added, &c., exactly in the same way as 2 and 3, the sum of the second being 7X2+5x3 5x7 and that of the first $x$t+$xt $x4 The rules for the addition, &c. of alge- braic fractions are exactly the same as in arithmetic ; for both the numerator 29 and denominator of every algebraic frac- tion stands either for a whole number or a fraction, and therefore the frac- tion itself is either of the same form as $ or-3, Nevertheless the student should attend to some examples of each opera- tion upon algebraic fractions, by way of practice in the previous operations. As the subject is not one which presents any peculiar difficulties, we shall now pass on to the subject of equations, con- cluding this article with a list of formule which it is highly desirable that the student should commit to memory before proceeding to any other part of the sub- ject. (a+ 6) + (a — d) = 2a (1) (a+ 6) —(a— 6) = 26 (2) a—(a—b)=86 (3) (a+ 6)? = a? + 2ab + 0? (4) (a—b=a@—2ab+ (5) (Qax + 6)? = 4a? x2? + 4abr + 6 (6) (a+ b)(a—-b=2— (7) (e+ a) (x +b) =2°+ (a+ b)x+aby (8) (t-a)(x«— 6b) = a2? —(a+b)xe+ab) a ma ‘ ee ‘y c ad+c ¢ ad—-c TAN AER, Gy Wa oo) a .¢ adtbe a c ad-—be Hitaedinlat DLE: + bayee Cau saan be wot a Gems a-Rape eer. ae eben ree 0") laugh oF dead (12) c a a Lipi cA ae (13) er oader G a a oats Wl Te Ane AE (14) d d ate a ec (15) b at Cle mata nd tLe the letters. For example, in the equation ‘Equations of the First Degree, We have already defined an equation, and have come to many equations of different sorts. But all of them had this character, that they did not depend upon the particular number which any letter stood for, but were equally true, what- ever numbers might be put in place of a— 1 atl the truth of the assertion made in this algebraical sentence is the same, whe- ther a be considered as representing 1, 2, 21, &c., or any other number or fraction whatever. The second side of this equation is, in fact, the result of the operation pointed out on the first side. On the first side you are directed to =a-J] ‘30 divide a? ~ 1 by a+ 15 the second side shews you the result of that division. An equation of this description is called an identical equation, because, in fact, a+a = 2a, 7a- 3a+h = 4a — 3b + 40, and The whole of the formulee at the end of the last article are examples of iden- tical equations. There is not one of them which is not true for all values which can be given to the letters which enter into them, provided only that what- ever a letter stands for in one part of an equation, it stands for the same in all the other parts. _ If we take, now, such an equation as a+1=8, we have an equation which is no longer true for every value which can be given to its algebraic quantities. -It is evident that the only number which a can represent consistently with this equation is 7, as any other supposition involves absurdity. This is a new spe- cies of equation, which can only exist in some particular case, which particular case can be found from the equation itself. The solution of every problem leads to such an equation, as will be shown hereafter, and, in the elements of algebra, this latter species of equation is of most importance. ~ In order to dis- tinguish them from identical equations, they are called equations of condition, because they cannot be true. when, the letters contained in them stand for any number whatever, and their very exist- ence makes a condition which the letters contained must fulfil. The solution of an equation of condition is the process of finding what number the letter must stand for in order that the equation may be true. Every such solution is a pro- cess of reasoning, which, setting out with supposing the truth of the equation, proceeds by self-evident steps, making use of the common rules of arithmetic and algebra. We- shall return to the subject of the solution of equations of condition, after showing, in a few in- stances, how we come to them in the solution of problems. In equations of condition, the quantity whose value is determined by the equation is usually represented by one of the last letters of the alphabet, and all others by some of the first. This distinction is necessary only for the beginner ; in time he must learn to drop it, and consider any letter as standing for a quantity known or un- known, according to the conditions of the problem. é | STUDY OF i its two sides are but different ways of | writing down the same number. This will be more clearly seen in the identical _ equations e = X Dies a. ‘In reducing problems to algebraical — equations no general rule can be given. ~ The problem is.some property of a number expressed in words by which ~ that number is to be found, and this pro- — perty must be written down as an equa- tion in the most convenient way. As examples of this, the reduction of the following problems into equations is © given ;— I. What number is that to which, if | 56 be added, the result will be 200 dimi- nished by twice that number ? Let x stand for the number which is to be found: Then 2+56=200 — 2a. If, instead of 56, 200, and 2, any other given numbers, a, 6, and c, are made use of in the same manner, the equation which determines 2 is xt+a=b-czx. II. Two couriers set out from the same place, the second of whom goes three miles an hour, and the first two. The first has been gone four hours, when the second is sent after him. How long will it be before he overtakes him? Let x be the number of hours which the second must travel to overtake the first. At the time when this event takes place, the first has been gone x-+4 hours, and will have travelled (x+4) 2, or 2x+8 miles. The second has been gone 2 hours, and will have travelled 3a miles. And, when the second overtakes the first, they have travelled exactly the same distance, and, therefore, 3L=22-+-8. If, instead of these numbers, the first goes a@ miles an hour, the second 6, and c hours elapse before the second is sent after the first, : | bx=ax-+ae. Four men, A, B, C, and D, built a ship which cost 26072, of which B paid twice as much as A, C paid as much as A and B, and Das much as C and B. What did each pay ? Suppose that A paid w pounds, then B paid2w... C paid e+ 2% or 38a). u's D paid 24+3a or 6046. MATHEMATICS, All together paid 7+227+3a% +5, or lla, therefore lla =2607, There are two cocks, from the first of which a cistern is filled in 12 hours, and the second in 15. How long would they be in filling it if both were opened toge- ther ? Let w be the number of hours which would elapse before it was filled. Then, since the first cock fills the cistern in 12 hours, in one hour it fills qs of it, in two hours =%, &c., and in & hours 5%. Similarly, in # hours, the second cock fills =. of the cistern. When. the two have exactly filled the cistern, the sum of these fractions must represent a whole or 1, and, therefore, ot aes : 121 16 If the times in which the two can fill the cistern are a and 6 hours, the equa- tion becomes =], prely Ba ° A person bought 8 yards of cloth for 31. 28., giving 9s. a yard for some of it and 7s. a yard for the rest; how much of each sort did he buy ? ‘ Let x be the number of yards at 7s. Then 72 is the number of shillings they cost. Also 8—@# is the number of yards. at 9s., and 8—2'9, or 72—9z; is the number of shillings they cost. And the ‘sum of these, or 7#+72—-92, is the whole price, which is 32. 2s., or 62 shil- lings, and, therefore, ; 7e@+72—90=62. . - These examples will be sufficient to show the method of reducing-a problem to an equation. Assuming a letter to stand for the unknown quantity, by means of this letter the same quantity must be found in two different forms, and these must be connected by the sign of equality. However the reduc- tion into equations of such problems as are usually given in the treatises on al- gebra rarely occurs in the applications of mathematics, The process is a use- ful exercise of ingenuity, but no student need give a great deal of time to it. Above all, let no one suppose, because he finds himself unable to reduce to equations the conundrums with which such books are usually filled, that, there- fore, he is not made for the study of mathematics, and should give it up. His future progress depends in no degree upon the facility with which he disco- 31 vers the equations of problems; we mean as fay as power of comprehending the subsequent sciences is concerned, He may never, perhaps, make any con- siderable step for himself, but, without doing this, he may derive all the benefits which the study of mathematics can afford, and even apply them extensively. There is nothing which discourages be- ginners more than the difficulty of re- ducing problems to equations, and yet, as respects its utility, if there be any- thing in the elements which may be dis- pensed with, it is this. We do not wish to depreciate its utility as an exercise for the mind, or to hinder all from attempt- ing to conquer the difficulties which pre- sent themselves; but to remind every one that, if he can read and understand all that is set before him, the essential benefit derived from. mathematical stu- dies will be gained, even though he should never make one step for himself in the solution of any problem. We return now to the solution of equations of condition. Of these there are various classes. Equations of the first’ degree, commonly called simple equations, are those which contain only the first power of the unknown quantity. Of this class are all the equations to which we have hitherto come in the so- lution of problems. The principle by which they are solved is, that two equal quantities may be increased or dimi- nished, multiplied, or divided by any quantity, and the results will be the same. In algebraical language, if a=6 atc=b+c6,a-c=6-cac= be and a c stated that any quantity may be removed from one side of the equation to the other, provided its sign be changed. This is nothing but an application of the principle just stated, as may be shown thus :—Let a + 6 — ¢ =d, add c to both quantities, then at+tb—c+e=d+cora+b=d-+e, Again subtract 6 from both quantities, thena+6-c—b=d—6, or a—c=d—b. Without always repeating the principle, it is derived from observation, that its effect is to remove quantities from one side of an equation to another, changing their sign at the same time. But the beginner should not use this rule until he is perfectly familiar with the manner of using the principle. He should, until he has mastered a good many examples, continue the operation at full length, instead of using the rule, which is an abridgment of it,. In fact it would be In every elementary book it is 32 better, and not more prolix, to abandon the received phraseology, and in the ex- ample just cited, instead of saying “ bring the term 0 to the other side of the equa- tion,” to say “subtract 6 from both sides,” and instead of saying “ bring c to the other side of the equation,” to say “add c to both sides.” Suppose we have the fractions 3, 4, and =°;. If we multiply them all by the product of the denominators 4 x7 x 14, or 392, all the products will be whole ; 3X392 1x 392 numbers, They will be Ata caat 5392 . : and rie and since 392 is measured by 4, 3 x 392 is also measured by 4, and 3X 392 is a whole number, and so on. But any common multiple of 4, 7, and 14 will serve as well. The least com- mon multiple wiil therefore be the most convenient to use for this purpose. The least common multiple of 4, 7, and 14, is 28, and if the three fractions be mul- 302 602 3 5 302. 30 Dap Genk. ut 3 1s : 210 30(3—2.2) 10 6 ¥ STUDY OF tiplied by 28, the results will be whole numbers. The same also applies to algebraic fractions. Thus => ~, and de | ap will become simple expressions, if they are multiplied by 6x dex bdf, or 6d ef. But the most.simple common multiple of 5, de, and bdf, is bdef, which should be used in preference to 2%d? ef. This being premised, we can now re- duce any equation which contains frac- tions to one which does not. For ex- ample, take the equation w 22 7 3-22 3 Prato. | dae If we multiply both these equal quan- tities by any other, the results will be equal. Wechoose, then, the least quan- tity, which will convert all the fractions into simple quantities, that is, the least common multiple of the denominators 3, 5,10, and 6, whichis 30. If we mul- tiply both equal quantities by 30, the equation becomes (1) 602. 60 x x, or 102, |G Sz X#, or 12a, &e,; so that we have 10% + 124 =21—5 (3 —22), (2) or 10% + 12% = 21 — (15 — 102), (3) or 102 + 122=21—15 + loa, (4) Beginners very commonly mistake this b-c+d-e af-b+c-dt+e process, and forget that the sign of sub- 2 = ——— rj = FATE Sey eyes traction, when it is written before a fraction, implies that the whole result ape 9 of A2 of the fraction is to be subtracted from a+b -+ — s aa ok the rest. As long as the denominator a+b até remains, there is no need to signify pee ano this by putting the numerator between a+ 6 — fae =" ana brackets, but when the denominator is a+b Ox: taken away, unless this be done, the sign of subtraction belongs to the first term of the numerator only, and not to the whole expression. The way to avoid this mistake would be to place in brackets the numerators of all fractions which have the negative sign before them, and not to remove those brackets until the operation of subtraction has been per- formed, as is done in equation (4), The following operations will afford exercise to the student, sufficient, perhaps, to enable him to avoid this error :— b-c+d-e _ aftb—-c+d-~e We can now proceed with the solution of the equation. Taking up the equation (4) which we have deduced from it, sub- tract 10 x from both sides, which gives 1074+12 ~—10 x=21-—15, or 12 x%=6: divide these equal quantities by 12, which : 12 6 1 ays} es —— = —. or X=. This is the» Page 12’ € only value which x can have so as to make the given equation true, or, as it is called, to satzsfy the equation. If, in- stead of x we substitute 4, we shall find — that— a+ = J Pe XR Til BR Ox b Wa aly sah — Cnet oie Yr- _- SS — me =? 3° 5 10 ee gs 10. 6’ MATHEMATICS. 4 4 : 1 i this we find to be true, since 7 4+ — js -— 5 Peg 22 10 6 60 equations of the first degree there is one unknown quantity, and all the others are known. These known quantities may be represented by letters, and, as we have said, the first letters of the al- phabet are commonly used for that aceha . abcehx _ Cc il 22 and and > aris In these 38 purpose, We will now take an equation of exactly the same form as the last, putting letters in place of numbers :— Sas) 2 —20 2 + ALAS | en! es Bae har) a c e Tier The solution of this equation is as follows: multiply both quantities by aceh, the most simple multiple of the de- nominators, it then becomes,— acdeh aceh(f—gez) eat 8 h t or, cehx + abehx = acdh — ace (f— ge), or, cehx + abehx = acdh — acef + acega. Subtract acegw from both sides, and it becomes cehx + abehu — acegx = acdh — acef or ceh + abeh— aceg\x = acdh — acef. Divide both sides by ceh + abeh — aceg, which gives acdh — acef x The steps of the process in the second case are exactly the same as in the first ; the same reasoning establishes them both, and the same errors are to be avoided in each. If from this we wish 8x5x7X6~83x5xX10X3 ie 5x 10K 6+ 3.x 2 X10 K 6 3K'S X LU x 2’ 552% 10-6 which is 4, the same as before. If in one equation there are two un- known quantities, the condition is not sufficient to fix the values of the two quantities; it connects them, never- theless, so that if one can be found the other can be found also. For example, the equation x+y = 8 ad- mits of an infinite number of solutions, for take @ to represent any whole num- ber or fraction less than 8, and let ¥ represent what w wants of 8, and this equation is satisfied. If we have another equation of condition existing between the same quantities, for example, 3a2—2y =4; this second equation by itself has an infinite number of solu- tions: to find them, y may be taken at pleasure, and a = avd Of all the solutions of the second equation, one only is a solution of the first ; thus there is only one value of 2 and y which satisfies both the equations, and the finding of these values is the solution of the equation. But there are some par- ticular cases in which every value of x and y which satisfies one of the equa- ~ ceh + ubeh— aceg. to find the solution of the equation first given, we must substitute 3 for a, 2 for 6, 5 for c, 7 for d, 10 for e, 3 for f, 2 for g, and 6 for h, which gives for the value of 4, 180 Ai) G area 360° 3’ >€ 5 54 E22 tions satisfies the other also; this hap- pens whenever one of the equations can be deduced from the other. For ex- ample, when 2+y=8, and 4 x7 — 29 =3—4y, the second of these is the same as 4%+4 y=3+29, or4 v+4y = 32, which necessarily follows from the first equation. If the solution of a problem should lead to two equations of this sort, it is a sign that the problem admits of an. infinite number of solutions, or is what is called an indeterminate problem. The solu- tion of equations of the second degree does not contain any peculiar difficulty ; we shall therefore proceed to the con- sideration of the isolated negative sign. CuHApPTEeR IX. On the Negative Sign, &c. Tr we wish to say that 8 is greater than 5 by the number 3, we write this equation 8 ~ 5=3. Also to say that a exceeds 6 by c, we use the equation a-—b=c. As long as some numbers whose value we know are subtracted from others equally known, ak is no 34 fear of our attempting to subtract the greater from the less; of our writing 3 —8 for example, instead of 8—3. But in prosecuting investigations in which letters occur, we are liable, sometimes from inattention, sometimes from igno- rance as to which is the greater of two quantities, or from misconception of some of the conditions of a problem, to re- verse the quantities in a subtraction, for example to write a — 6 where 6 is the greater of two quantities, instead of 6 —a. Had we done this with the sum of two quantities, it would have made no difference, becausea + 6 andbB+a are the same, but this is not the case with a — 6 and 6-a. For example, 8 — 3 is easily understood; 3 can be taken from 8 and the remainder is 5; but 3 — 8 is an impossibility, it requires you to take from 3 more than there is in 3, which is absurd. If such an ex- pression as 3 — 8 should be the answer to a problem, it would denote either that there was some absurdity inherent in the problem itself, or in the manner of putting it ito an equation. Never- theless, as such answers will oceur, the student must be aware what sort of mistakes give rise to them, and in what manner they affect the process of in- vestigation. We would recommend to the beginner to make experience his only guide in forming his notions ‘of these quantities, that is, to draw his rules from the ob- servation of many results, not from any theory. The difficulties which encom- pass the theory of the negative sign are explained at best in a manner which would embarrass him : probably he would not see the difficulties themselves ; too easy belief has always been the fault of young students in mathematics, and it is a great point gained to get them to start an objection. We shall observe the effect of this error in denoting a sub- traction on every species of investigation to which we have hitherto come, and shall deduce rules which the student will recollect are the results of experi- ence, not of abstract reasoning. The ex- tensions to which he will be led have ren- dered Algebra much more general than it was before, have made it competent to the solution of many, very many questions which it could not have touched, had they not been attended to. They do, in fact, constitute part of the groundwork of modern Algebra, and should be considered by the student Who is desirous of making his way into STUDY OF the depths of the science with the highest degree of attention. If he is well practised in the ordinary rules which have hitherto been explained, few diffi- culties can afterwards embarrass him, except those which arise from some con- fusion in the notions which he has formed upon this part of the subject. For brevity’s sake we hereafter use this phrase. Where the signs of every term in an expression are changed, it is said to have changed its form. Thus +a-—6 and+6- —-m— nh — : , which is of a different form m+n 2 from that in the sixth; but it must also be observed, that the first is after and the other before the moment when they are at A and B. In the fifth and sixth examples which differ in this, that the direction in which both are going is changed, since in the fifth they move towards one another, and in the sixth away from one another, the values of AH and BH im the one may be de- duced from those in the other by a change of form, both in m and 2, which gives the same values as before, But if 39 mand m change their forms in the ex- pression for the time, the value in the : : a sixth ease is —————, or — - -m—n m+rn Also the time in the fifth case is after the moment at which they are at A and B, and in the sixth case it is before. From these comparisons we deduce the following general conclusions :— 1. If we take the first case as astand- ard, we may, from the values which it gives, deduce those which hold good in all the other cases. Ifa second case be taken, and it is required to deduce an- swers to the second case from those of the first, this is done by changing the sign of all those quantities whose direc- tions are opposite in the second case to. what they are in the first, and if any an- swer should appear in a negative form, such as ma" ; ; ‘when m is less than 7, — which may be written thus — Liss n-~ m it is a sign that the quantity which it re- presents is different in direction in the first and second cases. If it bea right line measured from a given point in all the cases, such as AH, it is a sign that A Bi falls on the left in the second case, if it fell on the right in the first case, and the converse. If it be the time elapsed between the moment in which the cou- riers are at A and B and their meeting, it is a sign that the moment of meeting is before the other, in the second case, if it were after it in the first, and the con- verse. Wesee, then, that these six cases can be all contained in one if we apply this rule, and it is indifferent which of the cases is taken as the standard, pro- vided the corresponding alterations are made to determine answers to the rest. This detail has been entered into, in order that the student may establish, from his own experience, the general principle, which will conclude this part of the subject. Further illustration is contained in the following problem :— A workman receives a shillings a day for his labour, or a proportion of a shil- lings for any part of a day which he works. His expenses are 6 shillings every day, whether he works or no, and after m days he finds that he has gained c shillings. How many days did he work? Let w be that number of days, a being either whole or fractional, then for his work he receives az shillings, and during the m days his expenditure is 40 bm shillings, and since his gain is the difference between his receipts and ex- penditure ;— ax —bm=c hm+e Oe Now suppose that he had worked so httle as to lose ¢ shillings instead of gaining anything. The equation from which & is derived is now bm —axn=c which, when its form is changed, becomes ax-bm= —¢, an equation which only differs from the former in having —c¢ written instead ofc. The solution of the equation is mM —C aa. which only differs from the former in having —c instead of +c. It appears then that we may alter the solu- tion of a problem which proceeds upon the supposition of a gain into the solu- tion of one which supposes an equal loss, by changing the form of the expression which represents that gain; and also that ifthe answer to a problem which we have solved upon the supposition of a gain should happen to be negative, sup- pose it —c, we should have proceeded upon the supposition that there is a loss and should in that case have found a loss, c. When such principles as these have been established we have no occasion to correct an erroneous solution by recom- mencing the whole process, but we may, by means of the form of the answer, set the matter right at the end. The prin- ciple is, that a negative solution indicates that the nature of the answer is the very reverse of that which it was supposed to be in the solution; for example, if the solution supposes a line measured in feet in one direction. a negative answer, such as —c, indicates that c feet must be mea- sured in the opposite direction; if the answer was thought to be a number of days after a certain epoch, the solution shows that it is c days before that epoch ; if we supposed that A was to receive a cer- tain number of pounds, it denotes that he is to pay c pounds, and so on. In deducing this principle we have not made any sup- position as to what —c is; we have not asserted that it indicates the subtraction of ¢c from 0; we have derived the result from observation only, which taught us first to deduce rules for making that alteration in the result which arises from altering +¢ into —c at the commence- ment; and secondly, how to make the solution of one case of a problem serve STUDY OF | to determine those of all the others. By observation then the student must ac- quire his conviction of the truth of these rules, reserving all metaphysical discus- sion upon such quantitiesas +c and —c to a later stage, when he will be better prepared to understand the difficulties of the subject. We now proceed to another class of difficulties, which are generally, if possible, as much miscon- ceived by the beginner as the use of the negative sign. a 1 OM 3 Suppose its nu- merator to remain the same, but its — denominator to decrease, by which means the fraction itself is increased. For ex- Take any fraction 5 5 am ple>5 is greater than Tied thetwelfth part of 5 is greater than its twentieth ‘ 3 ee 23. 2 part. Similarly ao oe greater than a8 CW he G 2 &e. If, then, 6 be diminished more and more, the fraction 5 becomes greater and greater, and there is no limit to its pos- sible increase. To show this, suppose that bis a part of a, or that ba. Then i b ™m so as to be equal to any part of a, how- ever small, that is, so as to make m any a. : eee or—ism- Now since 6 may diminish a : . number, however great, — which is = m may be any number however great. This diminution of 6, and the consequent a b extent, which we may state in these words: As the quantity 6 becomes nearer and increase of —, may be carried on to any pane nearer to 0, the fraction 5 increases, and in the interval in which 6 passes from its first magnitude to 0, the fraction 7 passes from its first value through every pos- sible greater number. Now, suppose that the solution of a problem in its most f b cular case of that problem 6 is =0. We general form is —, but that in one parti- Pron: have, then instead of a solution, 92 sym- bol to which we have not hitherto given MATHEMATICS. ameaning. Totake an instance: return to the problem of the two couriers, and suppose that they move in the same direc- tion from C to D (Case first) at the same rate, or that m=n. Wefind that AH ma ma mM-nr m—nN the equation which produced this result . xX tx-a we find that it becomes — = Sor f= m m ma or On looking at —_— or x ~-a, which is impossible. On looking at the manner in which this equation was formed, we find that it was made on the supposition that A and B are together at some point, which in this case is also impossible, since if they move at the same rate, the same distance which separated them at one moment will separate them at any other, and they will never be to- gether, nor will they ever have been toge- ther on the other side of A. The con- clusion to be drawn is, that such an equation as = - indicates that the sup- position from which 2 was deduced can never hold good. Nevertheless in the common language of algebra it is said that they meet at an infinite distance, and that - is infinite. This phrase is one which in its literal meaning is an ab- surdity, since there is no such thing as an infinite number, that is a number which is greater than any other, because the mind can set no bounds to the mag- nitude of the numbers which it can con- ceive, and whatever number it can ima- gine, however great, it can imagine the next to it. But as the use of the phrase is very general, the only method is to attach a meaning which shall not in- volve absurdity or confusion of ideas. The phrase used is this. When c= acid == and is infinitely great. C= student should always recollect that this is an abbreviation of the following sen- The ; a tence. ‘* The fraction rae becomes greater and greater as ¢ approaches more and more near to 0; and if c, setting out from a certain value, should change gra- dually until it becomes ec~al to 4, the : a cs fraction ~—; setting out also from a cer- 9, | 41 tain value, will attain any magnitude however great, before c becomes equal to 6." That is, before a fraction can Gives i assume the form “ot it must increase without limit. The symbol 20 is used to denote such a fraction, or in general any quantity which increases without limit. The following equation will tend to elu- cidate the use of this symbol. In the problem of the two couriers, the equa- : ma L tion which gave the result “om ee m T-4 ™m which A ov) Gate =a, is eVi- dently impossible. Nevertheless, the larger x is taken the more near is this equation to the truth, as may be proved by dividing both sides by 2, when it be- ne ag comes 1 = 1 — ” , which is never exactly true. But the fraction = decreases as x increases, and by taking & sufficiently great may be reduced to any degree of smallness. For example, if it is required 1 10000000 of aunit, take x as great as 10000000a, that = should be as small as and the fraction becomes r a » 10000000a’ i 1 a eg eet EE Sio- Teoanaen 3 becomes smaller and smaller, the equation l= 1 — 4% be- Ay comes nearer and nearer the truth, which is expressed by saying that when a . 1 = Eb ~ ? or x = 2 — a, the solution is x=cc. Inthe solution of the problem of the two couriers this does not appear to hold good, since when m= and mM 2 oad the same distance a always separates them, and no travelling will bring them nearer together. To show what is meant by saying that the greater a is, the nearer willit be a solution of the problem, suppose them to have tra- yelled at the same rate to a great dis- A B | | 42 tance from C. They can never come together unless C A becomes equal to CB, or A coincides with B, which never happens, since the distance A B is always the same. But if we suppose that they have met, though an error always will arise from this false suppo- sition, it will become less and less as they travel further and further from C, For example, let C A = 10000000 AB, then the supposing that they have met, or that B and A coincide, or that BA=0, is an error which involves no more than of AC; and though AB is 10000000 always of the same numerical magnitude, it grows smaller and smaller in compa- rison with AC, as the latter grows greater and greater. Let us suppose now that in the pro- blem of the two couriers they move in the same direction at the same rate, as in the case we have just considered, but that moreover they set out from the same point, that is, leta=0. Itis now evident that they will always be together, that is, that any value of a2 whatever is an answer to the question. On looking at the value of AH, or — we find the numerator and denominator both equal to 0, and the value of AH appears . e 0 in the form e But from the problem we have found that one value cannot be assigned to A H, since every point of their course is a point where they are together. The solution of the following equation will further elucidate this. Let ax + by =c dxz--ey =f from which, by the common method of solution, we find ce te BP Saf acd he wea! > aed bd Now, let us suppose that ce = bf and ae = bd. Dividing the first of these by oft; Oe ifs od |g orcd=af. The values both of x and ce the second, we find ak a : ‘ 0 y in this case assumethe form Fit to find the cause of this we must return to the equations. If we divide the first of these by c, and the, second by /, we find that STUDY OF a pe d e 8 =y=1 bas: Yaw But the equations ce = bf and cd = af Dears give us ate Fads = s. that is, these two are, in fact, one and the same equa- tion repeated, from which, as has been explained before, an infinite number of values of x and y can be found; in fact, any value may be given to x provided be then found from the equation. We see that in these instances, when the — value of any quantity appears in the 0 A Pe form 7 that quantity admits of an infinite number of values, and this indicates that the conditions given to determine that quantity are not sufficient. But this is not the only cause of the appearance of a fraction in the form > Take the identical equation “i ae — $2 Ot b c(a—b) ¢ When a approaches towards 6, a+b approaches towards 2 4, and a? — 62 and a — b approach more and more nearly towards 0. If a = 6 the equation assumes this form: 35 a iy This may be explained thus: if we mul- tiply the numerator and denominator of the fraction a by a@— 6 (which does Aa-Ab Ba-BO If in the course of an investigation this has been done when the two quantities a and 6 are equal to one another, the frac- tion Oe br 2 Se he aka arte 10 B Dee Be Will appear in the notalter its value) it becomes form . But since the result would have A been B performed, this last fraction must be - used instead of the unmeaning form had that multiplication not been MATHEMATICS. 0 a&®— b2 ae fj j oa Se 0 Thus the fraction aes 0 peste eee) is the fraction oil ce (a—b) after its numerator and denominator have been multiplied by a— 5, and may be used in all cases except that in which 4= 6. When the form - occurs, the problem must be carefully examined in order to ascertain the reason. CHAPTER X. ‘Equations of the Second Degree. EVERY operation of algebra is connected 43 with another which is exactly opposite to it in its effects. Thus addition and subtraction, multiplication and division, __ are reverse operations, that is, what is ~ done by the one is undone by the other. Thus a-++ b— bisa, and i isa. Now in connexion with the raising of powers is a contrary operation called the ex- traction of roots. The term root is thus explained: We have seen that aa, or a’, is called the square of a; from which a is called the square root of a%. As 169 is called the square of 13, 13 is called the square root of 169. The following table will show how this phraseology is carried on, a is called the square root of a2, which is denoted by Ja? Gahat ei ue (ate Fe is Site a ° ° s &e. &e. [f 6 stand for a5, s/ } stands for a, and he foregoing table may be represented hus : li a2 = Bg = Nite a=b a= %/b, &e. The usual method of proceeding, is to each the student to extract the square ‘oot of any algebraical quantity imme- liately after the solution of equations of he first degree. We would rather re- -ommend him to omit this rule until he s acquainted with the solution of equa- ions of the second degree, except in the vases to which we now proceed. In irithmetic, it must be observed, that here are comparatively very few num- ers of which the square root can be xtracted. For example, 7 is not made ny the multiplication either of any whole jumber or fraction by itself. The first s evident; the second cannot be readily sroved to the beginner, but he may, by aking a number of instances, satisfy uimself of this, that no fraction which s really such, that is whose numerator s not measured by its denominator, vill give awhole number when mul- iplied by itself, thus 4 x § or 1° is nota whole number, and so on. The number 7, therefore, is neither the square of a vhole number, or of a fraction, and, pro- yerly speaking, has no square root, GUS LOGE, GE) GPa ciubays eRe ee fourth montiogiat. eo, se Ae . fifth root of a5, . . as ia &e. Nevertheless fractions can be found extremely near to 7, which have square roots, and this degree of nearness may be carried to any extent we please. Thus if required between 7 and 7 ysasds0000 could be found a fraction which has a square root, and the fraction in the last might be decreased to any extent what- ever, so that though we cannot find a fraction whose square is 7, we may nevertheless find one whose square is as near to 7 as we please. To take another example, if we multiply 1:4142 by itself the product is 1°99996164, which only differs from 2 by the very small fraction .00003826, so that the square of 1°4142 is very nearly 2, and fractions might be found where squares are still nearer to 2. Let us now suppose the following problem. A man buys a certain number of yards of stuff for two shillings, and the number of yards which he gets is exactly the number of shillings which he gives fora yard. Howmany yards does he buy? Let x be this number, then 2, : 2 Z3s the price of one yard, and x = rs or x22=2. This, from what we have said, is impossible, that is, there is no exact number of yards, or parts of yards, which will satisfy the conditions; never- theless 1°4142 yards will nearly do it, 1.4142136 still more nearly, and if the 44 problem were ever proposed in practice, there would be no difficulty in solving it with sufficient nearness for any purpose. A problem, therefore, whose solution contains a square root which cannot be extracted, may be rendered useful by approximation to the square root. Equations of the second degree, com- monly called quadratic equations, are those in which there is the second power, or square of an unknown quantity: such as z@ — 3 = 472 —- 15,a°+ 32= 2a2—g — 1, &e. By transposition of their terms, they may always be reduced to one of the following forms; o-+6=0 ax*—b=0 Ot aba + Ce 0 axt —- bx+c=0 az*+bx-c=0 ax? -bx-—c=0 For example, the two equations given above, are equivalent to 3a? — 12= 0, and x? — 4% —1=0, which agree in form with the second and last. In order to proceed to each of these equations, first take the equation 2% =a*, This equation is the same as 2% — a? = 0, or (x +a) (x — a) =0. Now, in order that the product of two or more quantities may be equal to nothing, it is sufficient that one of those quantities be nothing, and therefore a value of x may be derived from either of the following equations :— x-a=0 orz+a=0 the first of which gives x =a, and the second’ = —a. Toelucidate this, find x from the following equation :— 30+ a\(at+ v3) =(a® +ax) (a?+ar+22") develop this equation, and transpose all its terms on one side, when it becomes xt — 20x? +at+ 20° — 2az?=0 or (2? — a? )* — 2am (a? — a?) =0 or (v7 —a®) (x? — 2axv — a?) =0. This last equation is true when z — a?=0, or when a? = a’, which is true either when z = +4, or x = —a. If in the original equation + ais substituted in- stead of a, the result is 4a x 2a? =2a?x 4a? ; if — abe substituted instead of a, the result is 0=0, which show that +a and — a are both correct values of x. ‘We have here noticed, for the first time, an equation of condition, which is ca- pable of being solved by more than one value of x. We have found two, and shall find more when we can solve the equation x*-2ar—-a?=0, or w*—-2ar STUDY OF | =a", Every equation of the second degree, if it has one value of x, has a second, of which z* = a® is an in- stance, where 7 = -t a, in which by the’ double sign -b is meant, that either of them may be used at pleasure. We now, proceed to the solution of ax? — ba+c=0. In order to understand the nature of this equation, let us suppose that we take for # such a value, that ax? — bx+c, instead of being equal to 0, is equal to y, that is | y= ax? — bx--c* (1)? in which the value of y depends upon the value given to x, and changes when x changes. Let m be one of those quantities which, when substituted ine! stead of x, make ax? — bx -+-c equal to nothing, in which case mis called a root of the equation, ax? — br-+t+-c=0 (2); and it follows that am? — bm-+c¢ = 0 (3). Subtract (3) from (1), the result of which is y = a (a? — m2) —b(a —m) =x2- m\(ax-+m — b) Here y is evidently equal to 0, when v= m, as we might expect from the supposition which we made; but it is also nothing when a (w-++-m) - 6 = 0; there is, therefore, another value of a, for which y = 0; if we call this 2 we find it from the equation a (72+ m) -~b=0, Or? + 72S z 4 = z (4). In (3) substitute for } its value derived from (4), from which 6 = a(m—+-m) ; it then becomes . am? — am(n-+m)+e = 0, or ¢ —- amn = 0, ; wey) ode c which gives mn = — a (5). Substitute in (1) the values of 6 andeé derived from (4) and (5), which gives y = ax? — a(m-+-n)x-+-amn =a(a* -m+nx+mn). Now the second factor of this expression arises from multiplying together a — m = In the investigations which follow, a, b, and c are considered as having the sign which is marked before them, and no change of form is supposed tq take place, 4 } MATHEMATICS. ag As ‘and z — n: therefore, ° into the factors a, - mand & — , or y =a(e@—- mM) (x — Nn) (6). ax? —- br +e=a(x — Mm) (x — Nn). To take an example of this, let y = 4a? Tf we multiply a--m by atn, the péet. Here when wv = 1, y =4- } ba 5 +1 = 0, and therefore m=1. Ifwe Only difference between atm\ «+ n) divide 4a - 5a-+1 by a —1,* the a is RN Gommernnes 5 quotient (which is without remainder) and @ — m\x — n\is in the sign of the is 4a — 1, and therefore y= (@— 1) 42 — 1). This is also nothing when 4a —-1=0, @*- ba +e=a(e@—m) (x&—7n), . or when visi. Therefore = 41, and _ it follows,that y=4 (@ —1) (@ — 4), aresult coinciding aa*- ba+e=a(x-+m) (@-+ n). with that of (6). If, therefore, we can We now take the expression av? — find one of the values of x which satisfy ba —c. Ifthereis one value of w which the equation az®— b« +-c=0, wecan will make this quantity equal to 0, let find the other and can divide ax? — be+c this be m, and term which contains the first power of x If, therefore, Let y = ax? — bx —c Then 0 =am* — bm —c, © from which y = a (a2 — m*) — b (@ — m) = ©e—m(ax+m— Db) = x —m(ax+ an — bd). am — Let 2 be called n, or let am — 6 = an; then uv —m (ax + an) a(x —m) (@&+ 2). y Y _ As an example, it may be shown that | 3a? —e@~2 = 3(@— 1) (@ + 8). ' : Again, with regard to az? + ba—c, since x + m x —n only differs from x— m i + n in the sign of the term which contains the first power of a, itis evident that t if av? — ba — c=a(x4—™m) (& +n) ax® +-bxe —ce =a(x+m)(x& — n). Results ls to those of the first case may be obtained for all the others, and these results may be arranged in the following way. Inthe first and third, m is a quantity, which, when substituted for a, makes y = 0,and in the second and fourth mand 7 are the same as in the first and third. aa b c ) ist. y=ax®*—ba+c=a(ex—m)(4£—n) mt+n=—- mn= -. " a a em 2d. y=ar*+ bet+cHatetm) a@tn mtn= = m= +. . a a b Cc 8d. y= art — br ~c =a (@—m) (4+ 4%) ee Tee mn = —. . b Cn 4th. y =a? + baw —~c=a(e4+m) (c4#—n) lem eh age dia We must now inquire in what cases and observe that (2a — 5)? = 4a%x? a value can be found for wv, which will ~~ 4adbxr% + 5?. Multiply both sides of make y = 0 in these different expres- (1) by 4a, which gives sions, and in this consists the solution of day = 4a2a? —4abx + Aac (2). equations of the second degree. Let y = ax? — be +¢ Q), Add &? to the first two terms of the 46 second side of (2), and subtract it from the third, which will not alter the whole, and this gives day = 4a®x®? — 4aba + 68 + 4ac — 2 = (2axr — 6)? + 4ac — 62 = (3), Now it must be recollected that the square of any quantity is positive whe- ther that quantity is positive or ne- gative. This has been already suffi- ciently explained in saying that a change of the form of any expression does not change the form of its square. Common multiplication shows that c-—d! and d-d are the same thing; and, since one of these must be posi- tive, the other must be also positive. Whenever, therefore, we wish to say that a quantity is positive, it can be done by supposing it equal to the square 0 : b YR instead of x is put as = b+ Vb? —4ac STUDY OF of an algebraical quantity. In equation (3) there are three distinct cases to be considered. I. When 6? is greater than 4ae, that is, when 62 — 4ae is positive, lef 6° — 4dac = k®, which expresses the condition. Then 4ay = (2ax - 6)? =k? (4) and we determine those values of @, which make y=0, from the equation, — (2ax — 6)? —k? =0. We have already solved such an equation, and we find that 2ax-b= +8, where either sign may be taken. This shows that y or aa* —ba+c is equal to nothing either when ‘ = 7, 2a b—k b-V B= 4ac : ¥—— = —_____“= n, 2a 2a : the second values are formed by putting, instead of kits value 7 02—4ac. They are both positive quantities, because 2 being equal to 6®—4ac is-greater than 6?, and therefore & is greater than 3, (eer "eo av? — bx+c=a (x#—m) (x—n) =a (#— actual multiplication of the factors will show that this is an identical equation. II. When 2*, instead of being greater than 4ac, is equal to it, or when and therefore r and rae are ; both 2a positive. These are the quantities which we have called m and n in the former Investigations, and, therefore, b+ Af b2 — 2 (2 b—,/ el 2a . 2a g 6°— 4ac=0 and k=0. In this case the values of mand 7 are equal, each being Aes ‘ 2a et . 4 gz y=ax? —be+c=a (t—m) x—n\=a (e-=) a In this case y is said, in algebra, to be a perfect square, since its square root can be extracted, and is V a2 4) Arithmetically speaking, this would not be a perfect square unless a was a num- ber whose square root could be ex- tracted, but in algebra it is usual to call any quantity a perfect square with re- spect to any letter, which, when reduced, does not contain that letter under the sign 4/. This result is one which often occurs, and it must be recollected that when 62—4ac = 0, aa®—ba+c is a perfect square. ITI. When 2 is less than 4ac, or when l? — 4ac is negative and 4ac — b2 positive, let dac—b?=k2, and equation (3) becomes day =2ax — b |2+ k2, } In this case no value of x ean ever make y=0, for the equation vo? -+w?= indicates that v2 is equal to w? with a contrary sign, which cannot be, since all ‘squares have the same sign. The values of x are said, in this case, to be impossible, and it indicates that there ial something absurd or contradictory in the conditions of a problem which Ieads’ to such a result. Having found that whenever MATHEMATICS. ax? — be-+c=a (x—m) (“—n), it follows that az*+ba+c=a(a+m) (.v-+n). I. We know that when 2? is greater than 4ac, eX ali a ax?+bar+c=a (e+ Yo") (2+ Aj b ; b 2 # IJ. When ?=4ace, av?+ba+c=a (« as =) , and ¥ is a perfect squa IIT. When 82 is less than 4ac, ax?+ bx+c cannot be divided into Pieters. ; Now, let y=ax? — bx —c (1) As before, 4ay =4a? x* —4abx+ b? — 4ac — 6? nae = (2ax — b)? — (b2+4ac) oo ® ‘Let 62+ 4ac=k?. Then Aay = (2ax — b)2 — k?, (3) Therefore yi is 0 when 2axv— b\2=k2, or when 2a7—b= + &; b+k_ b+ JP tidac 2a 2a b—k _ b— AJB + 4ac 2a 2a Now, because 0? is less than b?+4ac, 6 is less than 4/ 4?+4ac, therefore 7 is a negative quantity. Leaving, for the present, the consideration of the negative quantity, we may decc decompose (3) into factors by means of the general formula pP—g=p—q pty, which gives 4ay=2ax-b-k 2axv—b+k Bus (tied 9 (gy teed 24 2a from which y or ; ema abo —o=a(a— VE ret? ) (e+ Jf PF 4ac ) 2a 2a Therefore, from what has been proved before, ®@+4ac+ Page ax +bxr—c=al £+ A b+ 4ae s) (2 A b?+4ac a) = 2a | 2a The following are some examples,’ of the truth of which the student should satisfy himself, both by reference to the one just established, and by actual multiplication :— 7 49 —2 Pees) Af AG Od 7 72 + 3.2 2 (ga OSs) (e—taain) = 2 (x-3) (w—1) a a \* 3a*7— 6a@ + 1=3 (Atv) (o-2"") Si v*—22@ + 22°'= 54 (L— 2) 221 + 9 221—9 ® Recollect that AN 24 = Vox = v6 x ng =x oN 6, ‘That is, m= n= Mia + br -c=a ( ign er az*—-be-c=4a These four ;cases may be contained in one, if we apply those rules for the change of signs which we have already established. ‘For example, the first side of (C) is made from that of (A), by changing the sign of c; the second side of (C) is made from ‘that of (A) in the same way. We have also seen the ‘suecessity of taking into account the ne- gative quantities which satisfy an equa- tion, as well as the positive ones; if we take these into account, each of the four forms of ax*+6x+e¢can be made —-b+ A b* — 4ac 2a mM = In the cases marked (B) (C) and (D), the results are b+ ./b® —4ac 2a — b+ t+ 4ac 2a b+ ,/0®+4ac 2a ~_ = m= m= and in ail the four eases the form of ax? + be+ ec which is used, is the same as the corresponding form of a(x—m) (~-n) and the following results may be easily obtained. In (A’), both m and x, if they exist at all, are negative. I say, if they exist at all, because it has been shown that if b2 dac is negative, the quantity axv* + ba + ecannot be divided into factors at all, since ./ l*-4ac is then no algebraical quantity, either po- sitive or negative. In (B’), both, if they exist at all, are positive. In (C’) there are always real values for m and n, since b® + 4ac¢ is always n= STUDY OF & the different results at which we have arrived, to whic Mion the student should take care to accustom himself, we have , ot vb lille ee) (e+! b+ fae wre («- be eee a —) (B) 5 eae) («- are | («- NO + act. a ) (et to ea aaa sesame, Uf hes ». Ag ee) (D) “ to nothing by two values of v For example, in (1), when ax? +ba+e=0 b- Jt? —4ac_ either 7+ aq 2a ‘ | b+ /6® — 4ac 2a chon i or z2+ If we call the values of x derived from) the equations m and 7, we find that —~b— Jb? —4ac Ty yee Se Oy al | b— Jb? —4ac dac y -b-— Ji?+4ac , b— J+ 4ac at AE (D1 positive ; one of these values is positive, and the other negative, and the negative, one is numerically the greatest. In (D’) there are also real values of m and m, one positive, and the other’ negative, of which the positive one is: numerically the greatest. Before pro- ceeding any further, we must notice an) ar". extension of a phrase which is usually adopted. The words greater and less, as applied to numbers, offer no diffi-| culty, and from them we deduce, that if. a be greater than 6,a—c is greater, than 6—c, as long as thave subtractions are possible, that j 1s, aS long as c can be taken both from a and 6. This is the only case which was considered when the rule.was made, but in extending the MATHEMATICS, | meaning of the word subtraction, and using the symbol — 3 to stand for 5-8, the principle that if @ be greater than 3, a — cis greater than 6 — c, leads to the following results. Since 6 is greater than 4,6 — 12 is greater than 4 — 12, or —6 is greater than—8; again 6=6 is greater than 4 —6, or 0 is greater than - 2. These results, particularly the Jast, are absurd, as has been noticed, if we continue to mean by the terms greater and less, nothing more than is usually meant by them in arithmetic; but in extending the meaning of one term, we must extend the meaning of all which are connected with it, and we are obliged to apply the terms greater and Jess in the following way. Of two algebraical quantities with the same or different signs, that one is the greater which, when both are connected with a number numerically greater than either of them, gives the greater result. Thus — 6 is said to be greater than — 8, be- cause 20 — 6 is greater than 20 — 8, 0 is greater than-—4, because 6+ 0 is greater than 6 —4; + 12 is greater — 30, because 40-++-12 is greater than 40 — 30. Nevertheless — 30 is said to be numerically greater than +12, be- ‘cause the number contained in the first is greater than that in the second. For this reason it was said, that in (C’), the hegative quantity was numerically greater than the positive, because any positive quantity is in algebra called greater than any negative one, even though the number contained in the first should be less than that in the second. In the same way — 14 is said to lie between + 3 and — 20, being less than the first and greater than the second. The advantage of these extensions is the same as that of others; the disadvan- tage attached to them, which it is not fair to disguise, is that, if used without proper caution, they lead the student into erroneous notions, which some elementary works, far from destroying, confirm, and even render necessary, by adopting these very notions as defi- nitions; as for example, when they say that a negative quantity is one which is less than nothing; as if there could be such a thing, the usual meaning of the word less being considered, and as if the student had an idea of a quantity less than nothing already in his mind, to which it was only necessary to give a fame. | | The product « -m wv — 7 is posi- | i 49 tive when x2 — m and aw — have the same, and negative when they have dif- ferent signs. This last can never happen except when x lies between m and 2, that is, when a is algebraically greater than the one, and less than the other. The following table will exhibit this, where different products are taken with various signs of m and n, and three values are given to x one after the other, the first of which is less than both m and m, the second between both, and the third greater than both. Value of the Product. Value of x. product“with its sign. x-4 27-7 et + 18 m=+4 + 5 - 2 t= 7 —- 10 +18 +102 —383 — 12 + 30 m= — 10 -—- 7 — 30 m=+3 + 4 +14 +2 e+12 ats + 9 HE = 2 —- 6 — 24 hw — 12 - | + il The student will see the reason of this, and perform a useful exercise in making two or three tables of this description for himself. The result is that z —m x — mis negative when 2 lies between mand m, is nothing when a is either equal to m or to 2, and positive when x is greater than both, or less than both. Consequently, @ (a —- m) (« — n) has the same sign as @ when @ is greater than both mand 2, or less than both, and a different sign from a when 2 lies between both. But whatever may be the signs of a, 6, and c, if there are two quantities m and 2, which make aw-+ bx -+e= a(t - m) (x — n), that is, if the equation ax*-++ be + c=0 has real roots, the expression ax?-|- bate always has the same sign as a for all values of a, except when a lies between these roots. It only remains to consider those cases in which av + bx + ¢ cannot be decom- posed into different factors, which hap- pens whenever 0° — 4ac is 0, or negative. In the first case when b’— 4ac = 0, we have ax* + bx-+-c=a (e+5) h 2 ax’ — be--c= a(« = ei E 50 and as these expressions are composed of factors, one of which is a square, and therefore positive, they have always the same sign as the other factor, which is a. When 0? — 4ac is negative, we have proved that if y=aa*tbxe-+c, day = (2av + 6)? + k2, where k? = 4ac — 62, and therefore day being the sum of two squares is always positive, that is, ax* + bx-+c has the same sign as a, whatever may be the value of x When c = 0, the expression becomes az®+ 62, or x (ax + 5b), which is nothing either when z= 0, or when aw + 6=0 and v= — °, the general expressions for m -b+ Jb and 2 become in this case ree 2u Ay RIE ? and eee which give the same results. When 6=0, the expression is re- duced to ax? + c= 0, which is nothing when @ = + Vs = es ‘which is not c possible, except when c and a have dif- ferent signs. In this case, that is, when the expression assumes the form az® —c, it is the same ds e(o-/%) (@ +/4). The same result might be deduced by making 6 =:0 in the general expressions for m and n. When a = 0, the expression is re- duced to 6x + c, which is made equal to nothing by one value of x only, that is — Fi If we take the general expres- sions for mand, and make a= 0 in . . -b Bb -— 4 them, that is, in esas ly a —b— Jf b?— 4ac and Gqute sa? We find as the results oni biop ; 5 and re These have been already explained. The first may either indi- cate that any value of «& will solve the problem which produced the equation ax? + bx +c=0, or that we have ap- plied a rule to a case which was not con- templated in its formation, and have thereby created a factor in the numerator and denominator of 2,which, in attempt- ing to apply the rule, becomes equal to - nothing. ‘The student is referred to the problem of the two couriers, solved in STUDY OF : the preceding part of this treatise. The ) latter is evidently the case here, because in returning to the original equation, we find it reduced to 62 +c= 0, which gives a rational value for a2, namely, C 25° The second value, or a | a which in algebraical language is called infinite, may indicate, that though there Deeds: a? which solves the equation, still that the greater the number which is taken for x, the more nearly is a second solution ob- tained. The use of these expressions is to point out the cases in which there is anything remarkable in the general problem; to the problem itself we must resort for further explanation. The importance of the investigations connected with the expression av? -F’ bx +c, can hardly be over-rated, at least to those students who pursue mas thematics to any extent. In the higher branches, great familiarity with these results is indispensable. The student is: therefore recommended not to proceed until he has completely mastered the de- tails here given, which have been hitherto too much neglected in English works on algebra. In solving equations of the second de- gree, we have obtained a new species of result, which indicates that the problem cannot be solved at all. We refer to those results which contain the square root of a negative quantity. We find that by multiplication the squares of ¢ —d and, of d — c are the same, both being c? — 2cd +d. Now either c—d or d= € is positive, and since they both have the: same square, it appears that the squares: of all quantities, whether positive or ne- gative, are positive. It is therefore absurd to suppose that there is any quantity which 2 can represent, and, which satisfies the equation a = — ay since that would be supposing that 2%, a positive quantity, is equal to the nega-. tive quantity — a%. The solution is then said to be impossible, and it will be easy) to show an instance in which such a re~. sult is obtained, and also to show that it arises from the absurdity of the problem. Let a number a be divided into any) two parts, one of which is greater than’ the half, and the other less. Call the: first of these si +a, then the second is no other value of w, except — must be < — HL, since the sum of beth | _ when the number a is halved. | a or — MATHEMATICS, _ partsmustbea, Multiply these parts to- part. of algebra is established which is gether, which gives . + x ) ( > —_ z), ; a\? eee os é or ( ‘) — x*. As x diminishes, this product increases, and is greatest of all when x = 0, that is, when the two parts, : : A See a a into which a is divided, are ZB and re or In this case the product of the parts. is > x > 2 oT and a number a can never be _ divided into two parts whose product is 2 greater than = This being premised, Suppose that we attempt to divide the. number a into two parts, whose product is d. Tet a be one of these parts, then a — xis the other, and their product is ax — x*. We have, therefore, aD — 28 = b a? —ar+b=0. If we solve this equation the two roots or are the two parts required, since from what we have proved of the expression x? —ax-+ bthe sum of the roots is a and their product 6. These roots are ey Gs pe 2 4 t . : ; ae which are impossible when eee b is negative, or when @ is greater than 2 a ; ; : | which agrees with what has just : been proved, viz. that no number is Capable of being divided into two parts 2 _ Whose product is greater than 7 We have shown the symbol / — a to be void of meaning, or rather self-contradictory and absurd. Never-, _ theless, by means of such symbols, a * The general expressions for m and n give ae er ee SE as the roots of #2 —axv +b =0. “ 52° of great utility. It depends upon the fact, which must be verified by experi- ence, that the common rules of algebra Imay be applied to these expressions ‘without leading to any false results. An appeal to experience of this nature ap- pears to be contrary to the first princi- ples laid down at the beginning of this work. We cannot deny that it is so in reality, but. it must be recollected that this is but a small and isolated part of an immense subject, to all other branches of which these principles apply in their fullest extent. There have not been wanting some to assert that these sym- bols may be used as rationally as any others, and that the results derived from them are as conclusive as any reasoning ' could make them. I leave the student to discuss this question as soon as he has acquired sufficient knowledge to under- stand the various arguments: at pre- sent, let him proceed with the subject as a part of the mechanism of algebra, on the assurance that by careful attention to the rules laid down, he can never be led to any incorrect result. The simple ~ rule is, apply all those rules to such ex- pressions as .J—a a+ /— 6, &e. which have been proved to hold good for such quantities as Ja at / 6, &e. Such expressions as the first of these are called imaginary, to distinguish them from the second, which are called real; and it must always be recollected that there is no quantity, either positive or. negative, which an imaginary expression can represent. ; It is usual to write such symbols as ff — 6 ina different form. To the equation —b= 6 x (—1) apply the rule derived from the equation /ay'= vax Ay, which gives Jf — b = /b x sf —1, of which the first factor Let Jb=c, then f—b=c ¥- 1. In this way all expressions may be so is real and the second imaginary. - arranged that ,/—1 shall be the only imaginary quantity which appears in them. Of this reduction the following are examples: E2 §2° J—24 = lao fe eg get yee anl =i “STUDY OF /2ab — a — B? = (a —)b) tf | V-a@x/-8 = afvl—xbsAf—1 = —ab. The following tables exhibit other applications of the rules: Ci i ch=a> sf —1, &e. em—3 = gi"™-3 sf - 1 c come 7 az c6 = — af, &e. cn" -2 =| = qs” =2 e=—g@/—] ci = — qi Wi fe &e. pated 2 Ligaen -tigl ag tale cD ng cs = as, Se. c4" = q4", The powers of such an expression as a,/ — 1 are therefore alternately real and imaginary, and are positive and negative in pairs. aa+bf—-1% =a —b+2ab/—-1 (a— bf —1) = a - B — 2abJ—1 (a+ Bie (a -ba ed =ar+ 6 atbJ/—-1 a&- | 2abJ/—1 pee fea were a? + 62 (atbJ—1) +d —1) =ace - bd + (ad4+ cb) J—1 Let the roots of the equation aa?-+ bx + c = 0 be impossible, that is, let b? — dac be negative and equal to — A®. Its roots, as derived from the rules esta- plished when 4? — 4ac was positive, are oxi) —. b2 a ES 2 be “Hb Rea Were he een, ile 2a 2a b eee k 6 k — “ioy pom 1s ist sithe Take either of these instead of a; for example, let 2a b ks i eh alee. aa anh 2) i aed sla eae ay — | — ——. Then ax? = 5a M 7 PUR FT Sly Ss A wes ii at aa ; Cacmu ¢ 2 Therefore, aa? + dva+c= — — a a be ; Pes), ei +c, in which, if 4ac — 8 be substi- tuted instead of k%, the result is 0. It appears, then, that the imaginary expres- sions which take the place of the roots when 62 — 4ac is negative, will, if the ordinary rules be applied, produce the same results as the roots. They are thence called imaginary roots, and we say that every equation of the second degree has two roots, either both real or both imaginary. It is generally true, that wherever an imaginary expression occurs, the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real. When an equation arises in which imaginary and real expressions occur together, such as a+6,/—l1 =c+ d f — 1, when all the terms are trans- ferred on one side, the part which is real and that which is imaginary must each of them be equal to nothing. The equation just given when its left side is transposed become a—c + 6—d)J—1 =0. Now, if 4 is not equal to d, let b—d =e; thn a—cteJ/—1=0, and ra | aca ; that is, an ima- ginary expression is equal to a real one, which is absurd. Therefore, 6 = d and the original equation is thereby reduced _ toa=c. This goes on the supposition MATHEMATICS. that a, b,c, andd arereal. If they are not so there is no necessary absurdity in — c—a@ mee Yc lia v e express that two possible quantities a and 6 are respectively equal to two others c and d, it may be done at once by the equation a+b /f—1=c-+d,/—1 The imaginary expression ,/—a and the negative expression — 6 have this resemblance, that either of them occur- ring as the solution of a problem in- dicates some inconsistency or absur- dity. As far as real meaning is concerned, both are equally imaginary, If, then, we wish to since 0 — ais as inconceivable as ,/—a. What, then, is the difference of signifi- eation? The following problems will elucidate this. A father is fifty-six, and his son twenty-nine years old: when will the father be twice as old as the son? Let this happen z years from the pre- sent time; then the age of the father will be 56 -+ a, and that of the son 29 + x; and therefore, 56 + x = 2(29-+ 7) = 58+ 2e, or x = — 2. This result is absurd; nevertheless, if in the equation we change the sign of x throughout it becomes 56—xv=58 — 22, or x=2. This equation is the one be- longing to the problem: a father is 56 and his son 29 years old; when was the father twice as old as the son? the answer to which is, two years ago. In this case the negative sign arises from too great a limitation in the terms of the problem, which should have de- manded how many years have elapsed or will elapse before the father is twice as old as his son? Again, suppose the problem had been ‘given in this last-mentioned way, In order to form an equation, it will be necessary either to suppose the event past or future. If of the two sup- positions we choose the wrong one, this error will be pointed out by the negative form of the result. In this case the Name of x. Square ofa - - - @v=aa S06... ks SE - a M=aaa Fourth Power - - ‘+ g#=aaaa Fifth Power - - - x=aaaaa 53 negative result will arise from a mistake in reducing the problem to an equation. In either case, however, the result may be interpreted, and a rational answer to the question may be given. This, how- ever, is not the case in a problem, the result of which is imaginary. Take the instance above solved, in which it is re- quired to divide a into two parts, whose product is 6. The resulting equation is w—art+b=0 a a? eee ae oe tet oY x ae fe the roots of which are imaginary when 2 6 is greater than > If we change the sign of x in the equation it becomes 2 an + b= 0 a at oan Sty [2s and the roots of the second are imagi nary, if those of the first are so. There is, then, this distinct difference between the negative and the imaginary result. When the answer to a problem is nega- tive, by changing the sign of w in the equation which produced that result, we may either discover an error in the method of forming that equation or show that the question of the problem is too limited, and may be extended so as to admit of a satisfactory answer. When the answer to a ‘problem is 1ma- ginary this is not the case. CHAPTER XI. On Roots in general, and Logarithms. Tue meaning of the terms square root, cube root, fourth root, &c. has already been defined. We now proceed to the difficulties attending the connexion of the roots of a with the powers of a. The following table will refresh the memory of the student with respect to the meaning of the terms :— Name of 2. Square Rootofa - eC Cube Root.... = ., Suppose it now required to multiply a» anda’. From (5) the first of these’ mg | is the same as axz, and the second is. | 4 ; the same as a. The product of these mg-+-np 29. | by the last case is a 7 , or Va"? But the! UL 95 edhe , and therefore — nq 2 q m P ed P a xat=artg (6) This is the same result as was ob- tained when the indices were whole numbers. The rule is ;—to multiply to- gether two powers of the same quantity, add the indices, and make the sum the index of the product. It follows in the same way that m a” amet eRe LECT RD "7 } mq—pn eek Gag ay or, to divide one power of a quantity by another, subtract the index of the di- visor from that of the dividend, and make the difference the index of the result. m\ P Suppose it required to find (*) mm ™m ™m™ m It is evident that a" xa” = Bi ge aie m\2/ | iam a”,or (,*) San 3m m\ 3 Similarly (i) = nie, ilies? Therefore (=) wt s 1 Again to find (,*) a or wri a” ,andso on. Let a J ite this be a. Then a = vs or Ee Ne bid Sa Be al Cr) =a",ora”’ =a". Therefore 1 ; ld q m () — ee, ante, - q my p Again to find () or Gry, 2g hm y nm ory nq’ and Apply the last two rules, and it appears a a 4 that () =U 5 ail i ee therefore (Gy apne a” 4, a=a’'; Ne ' And the rule is ;—to raise one power of a quantity to another power, multiply the indices of the two powers together, and make the product the index of the result, All these rules are exactly those which have been shown to hold good when the indices are whole numbers. But there still remains one remarkable extension, which will complete this subject. We have proved that whether m and m be whole or fractional numbers, a a a"~", The only cases which have been considered in forming this rule are those in which m is greater than , being the only ones in which the subtracted indi- cated is possible. If we apply the rule to any other case, a new symbol is pro- duced, which we proceed to consider. For example, suppose it required to find 3 ae, ar If we apply the rule, we find the MATHEMATICS, ~ result a3-7, or a-4, for which we have hitherto no meaning. As in former cases, we must apply other methods to the solution of this case, and when we have obtained a rational result, a~* may be used in future to stand for this result. 3 Now the fraction bes : 1 7 is the same as rh a a which is obtained by dividing both its numerator and denominator by a’. Bis : ”. is the rational result, for Therefore — which we have obtained a~* by applying a rule in too extensive a manner. Ne- vertheless, if.a~* be made to stand for 1 ] : ” and a-™ for al the rule will always give correct results, and the general rules for multiplication, division, and raising of powers remain the same as before. ail 1 For example, a~™ x a~" is — X —, or mm a” ——, which is wes or a7 (™ +) or qa” qn? vil ant ? 1 ane a7™ a” . a a~™-", Similarly , or —, is —, or Gat 5 ™ a my eng, 38 Again (abl is 1 1 ( m ye or RE OG As and so on. It a a a has before been shown that a° stands for 1 whenever it occurs in the solution ofa problem. We can now, therefore, assign a meaning to the expression a”, whether 7 be whole or fractional, posi- tive, negative, or nothing, and in all these cases the following rules hold good :— an x a" = a n (a) = The student can now understand the meaning of such an expression as 10°", where the index or exponent is a deci- 301 Since .301 is T000> this stands for "/ 10) *, an expression of which it would be impossible to calcu- late the value by any method which the student has hitherto been taught, but which may be shown by other processes to be yery nearly equal to 2. mal fraction. 57 Before proceeding to the practice of logarithmic calculations, the student should thoroughly understand the mean- ing of fractional and negative indices, and be familiar with the operations per- formed by means of them. He should work many examples of multiplication and division. in which they occur, for which he can have recourse to any ele- mentary work. The rules are the same as those to which he has been accus- tomed, substituting the addition, subtrac- tion, &c. of fractional indices, instead of these which are whole numbers. In order to make use of logarithms, he must, provide himself with a table. Hither of the following works may be recommended to him :— 1. Taylor's Logarithms, 2, Hutton’s Logarithms. Babbage; Logarithms of Num- bers, Callet; Logarithms of Sines, Co- sines, &c. 4. Bagay; Tables Astronomiques et Hydrographiques, The first and last of these are large works, calculated for the most accurate operations of spherical trigonometry and astronomy. The second and third are better suited to the ordinary student. For those who require a pocket volume, there are Lalande’s and Hassler’s Tables, the first published in Frange, the second in the United States. The definition, theory, and use of logarithms are fully given in the Trea- tise on Arithmetic and Algebra. The limits of this treatise will not allow us to enter into this subject. There is, however, one consideration connected with the tables, which, as it involves a principle of frequent application, it will be well to explain here. On looking into any table of logarithms it will be seen, that for a series of numbers the logarithms increase in arithmetical pro- gression, as far as the first seven places of decimals are concerned; that is, the difference between the successive loga- rithms continue the same. For example, the following is found from any tables: 3. Log. 41713 = 4.6202714 Log. 41714 = 4.6202818 Log. 41715 = 4.6202922 The difference of these successive loga- rithms and of almost all others in the same page is .0000104. Therefore in this the addition of 1 to the number gives an addition of .0000104 to the logarithm, Jt is a general rule that, 58 when one quantity depends for its value upon another,.as a logarithm does upon its number, or an algebraical expres- sion, such as 2% + # upon the letter or letters which it contains, if a very small addition be made to the value of one of these letters, in consequence of which the expression itself is increased or diminished, generally speaking, the in- crement * of the expression will be very nearly proportional to the imcrement of Bs iy ch =i Of xz? = .0001 z= .000001 © es _ Now quantities are compared, not by the actual difference which exists be- tween them, but by the number of times which one contains the other, and, of two quantities which are both very small, one may be very great as compared with the other. In the second example x? and x are both small fractions when compared with unity; nevertheless,, 2? is very great when compared with 2’, being 100,000 times its magnitude. This use of the words small and great some- times embarrasses the beginner ; never- theless, on consideration, it will appear to be very similar to the sense in which they are used in common life. We do not form our ideas of smallness or greatness from the actual numbers which are contained in a collection, but from the -proportion which the numbers bear to those which are usually found in similar collections. Thus of 1000 men we should say, if they lived in one village, that it was ex- tremely large; if they formed a regi- ment, that it was rather large; if an army, that it was utterly insignificant in point of numbers. Hence, in such an expression as Ah + Bh? + Ch?, we may, if / is very small, reject Bh? + CA8, as being very small compared with AA. An error will thus be committed, but a very small one only, and which becomes smaller as / becomes smaller. STUDY OF © the letter whose value is increased, and the more nearly so the smaller is the in- crement of the letter. We proceed to illustrate this. The product of two frae- tions, each of which is less than unity, is itself less than either of its factors. Therefore the square, cube, &c. of a fraction less than unity decrease, and the smaller the fraction is the more rapid is that decrease, as the following exam- ples will show :— Let x = .00001 xz = .0000000001 z= .000000000000001 &e. Let us take ‘any algebraical expres- sion, such as @ + a, and suppose that x is increased by a very small quantity fA. The. expression then becomes (v-- ht -+ (@ +h), or w+ a + (20 +-1)h-+ A*%. But it was a +a; _ therefore, in consequence of x receiving - the increment h, x*-++2@ has received © the increment 27 + 1)A-+ A2, for which — (2%-+1)h may be written, since fh is very small. since, if A were doubled, 27+ 1 A would This is proportional to h, ~ be doubled ; also, if the first were halved _ the second would be halved, &e. In general, if y is a quantity which contains — x, and if # be changed into x-Lh, y is changed into a quantity of the form y+ Ah + BA? + Chi + &c.; that is, y receives an increment of the form — Ah+BI+ Chs+-&e. If h be very © small, this may, without sensible error, be reduced to its first term, viz, Ah, which is proportional to 2. The general proof of this proposition belongs to a — higher department of mathematics ; ne- vertheless, the student may observe that it holds good in all the instances which — occur in the treatise on Arithmetic and — Algebra. For example (Tr. Arith. and Alg. page 89), ohn = xt ma" htm — gmn? f+. &e, mni— Here A = m2v"-1 B=m ad =a" + meh, nearly, Again (page 118), e* = 1 +h + ee + 1 ; } es a™-?, &e.; and if A be very small, 7 + h™ hs 2.3 i" &e. Therefore, e* x e* or e* +" = e* 1 ¢ he + &e. a ae ee aay * When any quantity is increased, the quantity by which it is increased is called its increment, MATHEMATICS. 59 And if A be very small, e+"= e+e" h, nearly; Again (page 119), log. (1--n’) = M (nm! — 4 n®#+4 0 — &e.)' To each side add log. , recollecting that log. v-+ log. (1 + n’) = log. (1+ 7’) = log. (v-+ an’),and let an! = h or n! = o equation becomes M M Making these substitutions, the Log. 7 +h =log. x+ a ST he + &e. If h is very small, log. ¢--h = log. # + ~ h. We can nowapply this to the logarithmic example with which we commenced this subject. It appears that Log. 41713 Log. (41713 +1) = 4.6202714 4,6202714 + .0000104 Log. (41713 +2) = 4.6202714 + .0000104 x 2. From which, and the considerations above-mentioned, | Log. (41713 LA) = log. 41713 + .0000104 x A, which is extremely near the truth, even when / is a much larger number, as the tables will show. Suppose, then, that the logarithm of 41713.27 is required. Here h = .27. It therefore only remains to calculate .0000104 x .27, and add the result, or as much of it as is contained in the first seven places of decimals, to the logarithm of 41713. This trouble is saved in the tables in the following manner. The difference of the succes- sive logarithms is written down, with the exception of the cyphers at the be- ginning, in the column marked;D or Diff, under which are registered the tenths of that difference, or as much of them as is contained in the first seven decimal places, increasing the seventh figure by 1 when the eighth is equal to or greater than 5, and omitting the cyphers to save room. From this table of tenths the table of hundredth parts may be made by striking off the last figure, making the usual change in the last but one, when the last is equal to or greater than 5, and placing an additional cypher. The logarithm of 41713.27 1s, therefore, obtained in the following man- ner :— Log. 41713 = 4.6202714 0000104 x .2 = .0000021 0000104 x .07=. .0000007 Log. 41713.27 = 4,6202742 This, when the useless cyphers and parts of the operation are omitted, is the pro- cess given in all the books of logarithms. If the logarithm of anumber containing more than seven significant figures be sought, for example 219034.717, recourse must be had toa table, in which the logarithms are carried to more than seven places of decimals. The fact is, that in the first seven places of decimals there is no difference between log. 219034.7 and log. 219034.717. For an excellent treatise on the practice of loga- rithms the reader may consult the pre- face to Babbage’s Table of Logarithms, CHAPTER XII. In this chapter we shall give the student some advice as to the manner in which he should prosecute his studies in alge- bra. The remaining parts of this sub- ject present a field infinite in its extent and in the variety of the applications which present themselves. By what- ever name the remaining parts of the subject may be called, even though the ideas on which they are based may be geometrical, still the mechanical pro- cesses are algebraical, and present con- tinual applications of the preceding rules and developments of the subjects already treated. This is the case in Trigono- metry, the application of Algebra to Geometry, the Differential Calculus, or Fluxions, &e. I. The first thing to be attended to in reading any algebraical treatise, is the gaining a perfect understanding of the different processes there exhibited, and of their connexion with one another. This cannot be attained by a mere read- ing of the book, however great the at- tention which may be given. It is im- possible, in a mathematical work, to fill up every process in the manner in which it must be filled up in the mind of the student before he can be said to have 60 completely mastered it. Many results must be given, of which the details are suppressed, such are the additions, mul- tiplications, extractions of the square roct, &c. with which the investigations abound. These must not be taken on trust by the student, but must be worked by his own pen, which must never be out of his hand while engaged in any algebraical process. The method which we recommend is, to write the whole of the symbolical part of each investigation, fillmg up the parts to which we have alluded, adding only so much verbal elucidation as is absolutely necessary to explain the connexion of the different steps, which will generally be much less than what is given in the book. This may appear an alarming labour to one who has not tried it, nevertheless we are convinced that it is by far the shortest method of proceeding, since the deli- berate consideration which the act of writing forces us to give, will prevent the confusion and difficulties which can- not fail to embarrass the beginner if he attempt, by mere perusal only, to under- stand new reasoning expressed in new language. If, while proceeding in this manner, any difficulty should oceur, it should be written at full length, and it will often happen that the misconception which occasioned the embarrassment will not stand the trial to which it is thus brought. Should there be still any matter of doubt which is not removed by attentive reconsideration, the student should proceed, first making a note of the point which he is unable to perceive. To this he should recur in his subse- quent progress, whenever he arrives at anything which appears to have any affinity, however remote, to the difficulty which stopped him, and thus he will fre- quently find himself in a condition to decypher what formerly appeared incom- prehensible. In reasoning purely geo- metrical, there is less necessity for com- l —— KX (l—27)= Again, a =1+2 Loga+ ; a =1+y Loga+ qty = | + ary Log a + Sieh. ae a Log a) 2 ¥2Loga -_—_— x+y\ Loga? STUDY OF mitting to writing the whole detail of the arguments, since the symbolical | language is more quickly understood, and the subject is in a great measure independent of the mechanism of ope- rations ; but, in the processes of algebra, there is no point on which so much de- | pends, or on which if becomes an in- structor more strongly to insist. II. On arriving at any new rule or process, the student should work a number of examples sufficient to prove © to himself that he understands and can | apply the rule or process in question. , Here a difficulty will occur, since there. are many of these in the books, to which - no examples are formally given. Never- theless, he may choose an example for himself, and his previous knowledge will suggest some method of proving whe- ther his result is true or not. For ex- ample, the development of a+a)\% will exercise him in the use of the binomial theorem ; when he has obtained the series which is equivalent to a+x\5, let him, in the same way, develop a+a3 3 the product of these, since 7 + £=3, 3 ought to be the same as the develop- ment ofate@, or as a°+ 3a2v7+ 3aa2+ 23, He may also try whether the develop- ment of a+, \! by the binomial theorem, gives the same result as is obtained by the extraction of the square root of a+x. Again, when any development is obtained, it should be seen whether the development possesses all the pro- perties of the expression from which it. has been derived. For example, : '% —F J is proved to be equivalent to the series 1+2+2°+2°+, &e., ad infinitum. This, when multiplied by 1—a, should give 1; when multiplied by 1—2®, should. give 1+2, because 1 —— x (l—2z*) = 1+2, &e. 1-2 riagl RALL ON x® Loga| 2.3 3 + &c. ad inf, +, &e. 2 + &e, MATHEMATICS. Now, since a® X a’ = a*t’, the pro- duct of the two first series should give the third. Many other instances of the same sort will suggest themselves, and a careful attention to them will confirm the demonstration of the several theo- rems, which, to a beginner, is often doubtful, on account of the generality of the reasoning. III. Whenever a demonstration ap- pears perplexed, on account of the num- ber and generality of the symbols, let some particular case be chosen, and let the same demonstration be applied. For example, if the binomial theorem should not appear sufficiently plain, the ame reasoning may be applied to the wot Agel 4. Bat ee to choose that of the third, or at most of the fourth degree, or both, on which to demonstrate all the properties of ex- pressions of this description. But in all these cases, when the particular instances have been treated, the general case should not be neglected, since the power of rea- soning upon expressions such as the one just given, in which all the terms cannot be written down, on account of their in- determinate number, must be exercised, before the student can proceed with any prospect of success to the higher branches of mathematics. IV. When any previous theorem is referred to, the reference should be made, and the student should satisfy himself that he has not forgotten its de- -monstration. If he finds that he has done so, he should not grudge the time necessary for its recovery. By so doing, he will avoid the necessity of reading over the subject again, and will obtain the additional advantage of being able to give to each part of the subject a time nearly proportional to its importance, whereas, by reading a book over and over again until he is master of it, he will not collect the more prominent parts, and will waste time upon unimportant details, from which even the best books are not free. The necessity for this conti- nual reference is particularly felt in the Elements of Geometry, where allusion is constantly made to preceding proposi- tions, and where many theorems are of no importance, considered as results, and are merely established in order to serve as the basis of future propositions. V. The student should not lose any opportunity of exercising himself in nu- merical calculation, and particularly in 61 ———- 2 expansion of 1 + 2%, or any other case, whichis there appliedto'l-+a7. Again, the general form of the product (7+), (7+ b),(a+c), &e.... containing n factors, will be made apparent by taking first two, then three, and four factors, before attempting to apply the reason- ing which establishes the form of the general product. The same applies par- ticularly to the theory of permutations and combinations, and to the doctrine of probabilities, which is so materially con- nected withit. Inthe theory of equa- tions it will be advisable at first, in- stead of taking the general equation of the form +La.+ M = 0, the use of the logarithmic tables. His power of applying” mathematics to ques- tic s of practical utility is in direct pro- portion to the facility which he possesses in computation. Though it is in plane and spherical trigonometry that the most direct numerical applications present themselves, nevertheless the elementary parts of algebra abound with useful practical questions. Such will be found resulting from the binomical theorem, the theory of logarithms, and that of continued fractions. ‘The first requisite in this branch of the rina gt is a perfect acquaintance with the arithmetic of de- cimal fractions; such a degree of aec- quaintance as can only be gained by a knowledge of the principles as well as of the rules which are deduced from them. From the imperfect manner in which arithmetic is usually taught, the student ought in most cases to recom- mence this study before proceeding to the practice of logarithms. Vie The oreatest difficulty, in fact al- most the onlyone of any importance which algebra offers to the reason, is the use of the isolated negative sign in such ex- pressions as — a, a~®, and the symbols which we have called imaginary. It is a remarkable fact, that the first elements of the mathematics, sciences which de- monstrate their results with more Cer- tainty than any others, contain difficulties which have been the subjects of dis- cussion for centuries. In geometry, for example, the theory of parallel lines has never yet been freed from the difficulty which presented itself to Euclid, and obliged him to assume, instead of prov- ing “the 12th axiom of his first book. Innumerable as have been the attempts 62 to elude or surmount this obstacle, no one has been more successful than another. The @iements of fluxions or the differential calculus, of mechanics, of optics, and of all the other sciences, in the same manner contain difficulties peculiar to themselves. These are not such as would suggest themselves to the beginner, who is usually embarrassed by the actual performance of the operations, and no ways perplexed by any doubts as to the foundations of the rules by which he is to work. It is the characteristic of a young student in the mathematical sciences, that he sees, or fancies that he sees, the truth of every result which can be stated in a few words, or ar- rived at by few and simple operations, while that which is long is always con- sidered by him as abstruse. Thus while he feels no embarrassment as to the meaning of the equation + a x —a = — a’, he considers the multiplication ofa” + a” by bm + b” as one of the difficulties of algebra. This arises, in our opinion, from the manner in which his previous studies are usually con- ducted. From his earliest infancy, he learns no fact from his own observation, he deduces no truth by the exercise of his own reason. Even the tables of arithmetic, which, with a little thought and calculation, he might construct for himself, are presented to him ready made, and it is considered sufficient to commit them to memory. Thus a habit of ex- amination is not formed, and the student comes to the science of algebra fully prepared to believe in the truth of any rule which is set before him, without other authority than the fact of finding it in the book to which he is recom- mended. It is no wonder, then, that he considers the difficulty of a process as proportional to that of remembering and applying the rule which is given, without taking into consideration the nature of the reasoning on which the rule was founded. Weare not advocates for stopping the progress of the student by entering fully into all the arguments for and against such questions, as the use of negative quantities, &c., which he could not understand, and which are inconclusive on both sides; but he might be made aware that a difficulty does exist, the nature of which might be pointed out to, him, and he might then, by the consideration of a sufficient number of examples, treated separately, acquire confidence in the results to which STUDY OF the rules lead. Whatever may be thought of this method, it must be better than an unsupported rule, such as is given in many works on algebra. It may perhaps be objected that this is induction, a species of reasoning which | is foreign to the usually received notions of mathematics. To this it may be answered, that inductive reasoning is of as frequent occurrence in the sciences as any other. It is certain that most great discoveries have been made by means of it; and the mathematician knows that one of his most powerful engines of demonstration is that peculiar species _ of induction which proves many general truths by demonstrating that, if the’ theorem be true in one case, it is true for the succeeding one. But the be- ginner is obliged to content himself with’ a less rigorous species of proof, though equally conclusive, as far as moral cer- tainty is concerned. Unable to grasp the generalizations with which the more advanced student is familiar, he must satisfy himself of the truth of general theorems by observing a number of particular simple instances which he is able to comprehend. For example, we would ask any one who has gone over this ground, whether he derived more certainty as to the truth of the binomial theorem from the general demonstration af indeed he was suffered to see it so early in his career), or from observation of its truth in the particular cases of the 2 3 development ofa +6, a+o0, &e, substantiated by ordinary multiplication. We believe firmly, that to the mass of young students, general demonstrations afford no conviction whatever; and that the same may be said of almost every species of mathematical reasoning, when it is entirely new. We have before ob- served, that it is necessary to learn to reason; and in no ease is the assertion more completely verified than in the study of algebra. It was probably the ex- perience of the inutility of general de- monstrations to the very young student that caused the abandonment of rea- soning which prevailed so much in English works on elementary mathe-. matics. Rules which the student could follow in practice supplied the place of arguments which he could not, and no pains appear to have been taken to adopt a middle course, by suiting the nature of the proof to the student’s capacity. The objection to this appears. | | ; ; , ' MATHEMATICS, | to have been the necessity which arose for departing from the appearance of rigorous demonstration. This was the ery of those, who not having seized the spirit of the processes which they fol- lowed, placed the force of the reasoning in the forms. To such the authority of great names is a strong argument; we will therefore cite the words of LAPLACE on this subject. ** Newton extended to fractional and negative powers the analytical expression which he had found for whole and_po- sitive ones. You see in this extension one of the great advantages of algebraic ‘language, which expresses truths much more general than those which were at first contemplated, so that by making the extensions of which it admits, there arises a multitude of new truths out of formulz which were founded upon very limited suppositions. At first, people were afraid to admit the general con- sequences with which analytical for- mule furnished them; but a great number of examples having verified ~ them, we now, without fear, yield our- selves to the guidance of analysis through all the consequences to which it leads us, and the most happy dis- coveries have sprung from the bold- ness. We must observe, however, that precautions should be taken to avoid giving to formule a greater extension than they really admit, and that it is always well to demonstrate rigorously the results which are obtained.” We have observed, that beginners are not disposed .to quarrel with a rule which is easy in practice, and verified by examples, on account of difficulties which occur in its establishment. The early history of the sciences presents occasion for the same remark. In the work of Diophantus, the first Greek writer on algebra, we find a principle equivalent to the equations + a x — 6 =-a b, and -ax —b= +48, admitted as an axiom, without proof or difficulty. In the Hindoo works on algebra, and the Persian commentators upon them, the same thing takes place. It appears, that struck with the practical utility of the rule, and certain by in- duction of its truth, they did not scruple to avail themselves of it. A more cul- tivated age, possessed of many formule whose developments presented striking examples of an universality in algebraic language not contemplated by its framers, set itself to inquire more closely 63 into the first principles of the science. Long and still unfinished discussions have been the result, but the progress of nations has exhibited throughout a strong resemblance to that of indivi- duals. VII. The student should make for himself a syllabus of results only, unac- companied by any demonstration. It is essential to acquire a correct memory for algebraical formule, which will save much time and labour in the higher de- partments of the science. Such a syl- labus will be a great assistance in this respect, and care should be taken that it contain only the most useful and most prominent formule. Whenever that can be done, the student should have recourse to the system of tabulation, of which he will have seen several examples in this treatise. In this way he should write the various forms which the roots of the equation ax? + bx +c=0 assume, accord- ing to the signs of a, 6, and c, &c. Both the preceptor and the pupil, but espe- cially the former, will derive great ad- vantage from the perusal of Lacrova, Essais sur l Ensergnement en général et sur celui des Mathématiques en parti- culter, Condillac, La Langue des Calculs, and the various articles on the elements of algebra in the French Encyclopedia, which are for the most part written by D’Alembert. The reader will here find the first principles of algebra, developed and elucidated in a masterly manner. A great collection of examples will be found in most elementary works, but particularly in Hirsch, Sammlung von Beispielen, &c. translated into English under the title of Self Hxaminations in Algebra, §c., London, Black, Young, and Young, 1825. The student who desires to carry his algebraical studies farther than usual, and to make them the stepping-stone to a knowledge of the higher mathematics, should be ac- quainted with the French language. A knowledge of this, sufficient to enable him to read the simple and easy style in which the writers of that nation treat the first principles of every subject, may be acquired in a short time. When that is done, we recommend to the student the algebra of M. Bourdon, a work of emi- nent merit, though of some difficulty to the English student, and requiring some previous habits of algebraical reasoning. VIII. The height to which algebraical studies should be carried must depend upon the purpose to which they are to 64 be applied. For the ordinary purposes of practical mathematics, algebra is principally useful as the guide to trigo- nometry, logarithms, and the solution of equations. Much and profound study is not therefore requisite; the student should pay great attention to all nume- rical processes, and particularly to the methods of approximation which he will find in all the books. His principal in- strument is the table of logarithms, of which he should secure a knowledge -both theoretical and practical. The course which should be adopted pre- paratory to proceeding to the higher branches of mathematics is different. It is still of great importance that the student should be well acquainted with numerical applications; nevertheless, he may omit with advantage many de- tails relative to the obtaining of approxi- STUDY OF MATHEMATICS. mative numerical results, particularly in the theory of equations of higher de- grees than the second. Instead of oc- cupying himself upon these, he should proceed to the application of algebra to geometry, and afterwards to the differential calculus. When a com- petent knowledge of these has been ob- tained, he may then revert to the sub- jects which he has neglected, giving them more or less attention according to his own opinion of the use which he is likely to have for them. This applies particularly to the theory of equations, which abounds with processes of which very few students will afterwards find the necessity. We shall proceed in the next number to the difficulties which arise in the study of Geometry and Trigonometry. STUDY OF MATHEMATICS. 65 CHAPTER XIII. On the Definitions of Geometry. In this treatise on the difficulties of Geo- metry and Trigonometry, we propose, as in the former part of the work, to touch on those points only which, from novelty in their principle, are found to present difficulties to the student, and which are frequently not sufficiently dwelt upon in elementary works. Per- haps it may be asserted, that there are no difficulties in geometry which are likely to place a serious obstacle in the way of an intelligent beginner, except the temporary embarrassment which always attends the commencement of a new study; that, for example, there is nothing in the elements of pure geometry com- parable, in point of complexity, to the theory of the negative sign, of fractional indices, or of the decomposition of an ex- ‘pression of the second degree into fac- tors. This may be true; and were it only necessary to study the elements of this science for themselves, without re- ference to their application, by means of algebra, to higher branches of knowledge, we should not have thought it necessary to call the attention of our readers to the points which we shall proceed to place before them. But while there is a higher study in which elementary ideas, simple enough in their first form, are’so gene- ralized as to become difficult, it will be an assistance to the beginner who intends to proceed through a wider course of pure mathematics than forms part of common education, if his attention is early di- rected, in a manner which he can com- prehend, to future extensions of what is before him. - The reason why geometry is not so difficult as aigebra, is to be found in the less general nature of the symbols em- ployed. In algebraa general proposition respecting numbers is to be proved. Letters are taken which may represent any of the numbers in question, and the course of the demonstration, far from making any use of a particular case, does not even allow that any reasoning, however general in its nature, is con- clusive, unless the symbols are as general as the arguments. We do not say that it would be contrary to good logic to form general conclusions from reasoning on one particular case, when it is evident that the same considerations might be applied to any other, but only that very great caution, more than a beginner can see the value of, would be requisite in deducing the conclusion. There occurs also a mixture of general and particular propositions, and the latter are liable to be mistaken for the former. In geometry on the contrary, at least in the elemen- tary parts, any proposition may be safely demonstrated by reasonings on any one particular example. For though in proving aproperty of a triangle many truths regarding that triangle may be asserted as having been proved before, none are brought forward which are not general, that is, true for allinstances ofthe same kind. It also affords some facility that the results of elementary geometry are in many cases sufficiently evident of themselves to the eye; for instance, that two sides of atriangle are greater than the third, whereas in algebra many rudimentary propositions derive no evi- dence from the senses; for example, that a® — 6° is always divisible without re- mainder by a — 6, The definitions of the simple terms point, line, and surface have given rise to much discussion. But the difficulties which attend them are not of a nature to embarrass the beginner, provided he will rest content with the notions which he has already derived from observation. No explanation can make these terms more intelligible. To them may be added the words straight line, which cannot be mistaken for one moment, unless it be by means of the attempt to explain them by saying that a straight line is ‘ that which lies evenly between its extreme points.’ The line and surface are distinct species of magnitude, as much so as the yard and the acre. The first is no part of the second, that is, no number of lines can make a surface. When therefore a surface is divided into two parts by a line, the dividing line is not to be con- sidered as forming a part of either. That the idea of the line or boundary necessarily enters into the notion of the division is very true; but if we conceive the line abstracted, and thus get rid of the idea of division, neither or i is 66 increased or diminished, which is what we mean when we say that the line is not apart of the surface. The same con- siderations apply to'a point, considered as the boundary of the divisions of a line. The beginner may perhaps imagine that a line is made up of points, that is, that every line is the sum of a number of points, a surface the sum of a number of lines, andso on. This arises from the fact, that the things which we draw on paper as the representatives of lines and points, have in reality three dimen- sions, two of which, length and breadth, are perfectly visible. Thus the point, such as we are obliged to represent it, in order to make its position visible, is in reality a part of our line, and our points, if sufficiently multiplied in number and placed side by side, would compose a line of any length whatever. But taking the mathematical definition of a point, which denies it all magnitude, either in length, breadth, or thickness, and of a line, which is asserted to pos- sess length only without breadth or thickness, it is easy to shew that a point is no part of a line, by making it appear that the shortest line can be cut in as many points as the longest, which may be done in the following manner. Let AB be any straight line, from the ends of which, A and B, draw A. Qh F f c B two lines, A F and C B, parallel to one another. Consider AF as produced without limit, and in CB take any point C, from which draw lines CE, CF, &c., to different points in AF, It is evident that for each point E in AF there is a distinct point in AB, viz., the intersection of C E with A B ;— for, were it possible that two points, E and F in AF, could be thus connected with the same point of A B, it is evident that two straight lines would enclose a space, viz., the lines C E and CF, which both pass through C, and would, were our supposition correct, also pass through the same pointin AB. There can then be taken as many points in the finite or bounded line AB as in the indefinitely extended line A F. The next definition which we shall consider is that of a plane surface. The word plane or flat is as hard to STUDY OF define, without referenee to any thing but the idea we have of it, as it is easy to understand. Nevertheless the prac- tical method of ascertaining whether or no a surface is plane, will furnish a definition, not such, indeed, as to render the nature of a plane surface more evident, but which will serve, in a mathematical point of view, as a basis on which to rest the propositions of solid geometry. If the edge of a ruler, known to be perfectly straight, coincides with a surface throughout its whole length, in whatever direction it may be placed upon that surface, we conclude that the surface is plane. definition of a plane surface is that in which, any two points being taken, the straight line joining these points lies wholly upon the surface. Two straight lines have a relation to one another independent altogether of their length. This we commonly express (for among the most common ideas are found the germ of every geometrical theory) by saying that they are in the same or different directions. By the direction of the needle we ascertain the direction in which to proceed at sea, and by the direction in which the hands of a clock are placed we tell the hour. It remains to reduce this common notion to a more precise form. Suppose a straight line OA to be given in magnitude and position, and to remain fixed while another line OB, at first coincident with O A, is made to move round O A, so as continually to vary its direction with respect to O A. The process of opening a pair of com- passes will furnish an illustration of this, but the two lines need not be equal to one another. In this case the opening made by the two will continually in- crease, and this opening is a species of magnitude, since one opening may be compared with another, so as to ascertain which of the two is the greater. Thus if the fig. CPD be removed from its place, without any other change, so that the point P may fall on O, and the line P C may lie upon and become a part of O A, or OA of PC, according to which is the longer of the two, then if the opening C P Dis the same as the opening A OB, PD will lie upon A B atthe same time a ae Hence the — | -as PC lies upon OA. Butif PD does not then lie upon O B, but falls between OB and OA, the opening C P D is less than the opening A OB, and if PD does not fall between OA and OB, or on OB, the opening C P D is greater than the opening BOA. To this species of magnitude, the opening of two lines, the name of angle is given, that is B O is said to make anangle withOA. The difficulty here arises from this mag- nitude being one, the measure of which has seldom fallen under observation of those who begin geometry. Every one has measured one line by means of another, and has thus made a number the representative of a length; but few, at this period of their studies, have been accustomed to the consideration, that one opening may be contained a certain number of times in another, or may be a certain fraction ofanother. Nevertheless we may find measures of this new species of magnitude either by means of time, length, or number. One magnitude is said to be a mea- sure of another, when, if the first be doubled, trebled, halved, &c., the second is doubled, trebled, or halved, &c.; that is, when any fraction or multiple of the first corresponds to the same fraction or multiple of the second in the same man- ner as the first does to the second. The two quantities need not be of the same kind: thus, in the barometer the height of the mercury (a length) measures the pressure of the atmosphere (a weight) ; for if the barometer which yesterday stood at 28 inches, to-day stand at 29 inches, in which case the height of yes- terday is increased by its 28th part, we know that the atmospheric pressure of yesterday is increased by its 28th part to-day. Again, in a watch, the number of hours elapsed since twelve o'clock is measured by the angle which a hand makes with the position it occupied at twelve o'clock. In the spring balances a weight is measured by an angle, and Cc ¥F aN , B if e A G IL many other_ similar instances might be given, MATHEMATICS. 67 This being premised, suppose a line which moves round another as just de- scribed, to move uniformly, that is, to describe equal openings or angles in equal times. Suppose the line OA to move completely round, so as to reas- sume its first position in twenty-four hours. Then in twelve hours the mov- ing line will be in the position OB, in six hours it will be in OC, and in eighteen hours in OD. The line O C is that which makes equal angles with O A and OB, and is said to be at right angles, or perpendicular to O A and OB. Again, OA and OB which are in the same right line, but on opposite sides of the point O, evidently make an opening or angle which is equal to the sum of the angles A OC and C OB, or equal to two right angles. A line may also be said to make with itself an open- ing equal to four right angles, since after revolving through four right an- gles, the moving line reassumes its ori- ginal position. We may even carry this notion further : for if the moving line be in the position O E when P hours have elapsed, it will recover that position after every twenty-four hours, that is, for every additional four right angles de- scribed; so that the angle A O EF is equally well represented by any of the following angles: 4 right angles + AOE 8 right angles + AOK 12 right angles + AOE &c. &e. &e. These formule which suppose an open- ing greater than any apparent opening, and which take in and represent the fact that the moving line has attained its position for the second, third, fourth, &e., time, since the commencement of the motion, are not of any use in ele- mentary geometry; but as they play an important part im the application of algebra to the theory of angles, we have thought it right to mention them here. ; It is plain also that we may conceive the line OE to make two, openings or angles with the original position OA 1. that through which it has moved to recede from OA; 2. that through which it must move to reach O A again. The first (in the position in which we have placed O A) is what is called in geome- try the angle AO E; the second is more simply described as composed of the openings or angles EOC, COB, BOD, DOA, and is not used excent a the ap- 68 plication of algebra above mentioned. Of the two angles just alluded to, one must be less than two right angles, and the second ‘greater; the first is the one usually referred to. It is plain that the angle or opening made by two lines does not depend upon their length but upon their position ; if either be shortened or lengthened, the angle still remains the same; and if while the angle increases or decreases one of the straight lines containing it is diminished, the angle so contained may have a definite and given magnitude at the moment when the straight line dis- appears altogether and becomes no- thing. For example, take two points of any curve A B, and join A and B by a straight line. Let the point B move to- wards A; it is evident that the angle made by the moving line with A B in- creases continually, while as much of one of the lines containing it as is inter- cepted by the curve, diminishes without limit. When this intercepted part dis- appears entirely, the line in which it would have lain had it had any length, has reached the line AG, which is called the tangent of the curve. In elementary geometry two equal angles lying on different sides of a line, such as AOE, A OH, would be con- sidered as the same. In the applica- tion of algebra, they would be consi- dered as having different signs, for rea- sons stated at length in pages 37, &c., of the first part of this Treatise. Itis also common in the latter branch of the sci- ence, to measure angles in one direction only; for example, in the same figure, the angles made by OE, OF, OG, and O H, if measured upwards from O A, would be the openings through which a line must move in the same direction from OA, to attain those positions ; and the second, third, and fourth angles would be greater than one, two, and three right angles respectively. We proceed to the method of reason- ing in geometry, or rather to the method of reasoning in general, since there is, or ought to be, no essential difference between the manner of deducing results from first principles, in any science. STUDY OF CuapTer XIV. On Geometrical Reasoning. Ir is evident that all reasoning, of what | | | form soever, can be reduced at last to anumber of simple propositions or as- | sertions; each of which, if it be not self-evident, depends upon those which | have preceded it. Every assertion can | be divided into three distinct parts. Thus the phrase ‘all right angles are equal,’ consists of—1, the subject spoken of, viz. right angles, which is here spoken of universally, since every nght angle is a part of the subject;—2, the copula, or manner in which the two are joined together, which is generally the verb is, or is equal to, and can always be reduced to one or the other: in this case the copula is affirmative ;—3, the. predicate, or thing asserted of the sub- ject, viz., equal angles. The phrase, thus divided, stands as written below, and is called universally affirmative. The second is called a particular affirma- tive proposition; the third a universal ne- gative; the fourth a particular negative. All right angles are equal, (magni- tudes). Some (figures). No circle is convex to its diameter. Some triangles are not equilateral, (figures). Many assertions appear in a form which, at first sight, cannot be reduced to one of the preceding: the following are instances of the change which it is necessary to make in them. Parallel lines never meet, or parallel lines are lines which never meet. The angles at the base ofan isosceles triangle are equal, or an isosceles triangle is a tri- angle having the angles at the base equal. The different species of assertions, and the arguments which are com- pounded of them, may be distinctly con- ceived by referring them all to one species of subject and predicate. Since every assertion, generally speaking, in- cludes a number of individual cases in its subject, let the points of a circle be the subject and those of a triangle the predicate. These figures being drawn, the four species of assertions just alluded to are as follows :— 1. Every point of the circle is a point of the triangle, or the circle is contained in the triangle. 2. Some points of the circle are points of the triangle, or part of the circle is contained in the triangle. triangles are equilateral, ~MATHEMATICS. 8. No point of the circle is a point of the triangle, or the circle, is entirely without the triangle.’ 4. Some points of the circle are not points of the triangle, or part of the circle is outside the triangle. On these we observe that the second follows from the first, as also the fourth _ from the third, since that which is true of all is true of some or any; while the first and third do not follow from the second and fourth, necessarily, since that which is true of some only need not be true ofall, Again, the second and fourth are not necessarily inconsistent with each other for the same reason. Also two of these assertions must be true and the others untrue. The first and the third are - ealled contraries, while the first and fourth, and the second and third are contradictory. The converse of a pro- position is made by changing the pre- dicate into the subject, and the subject into the predicate. No mistake is more common than confounding together a proposition and its converse, the ten- dency to which is rather increased in those who begin geometry, by the number of propositions which they find, the converses of which are true. Thus all the definitions are necessarily conversely true, since the identity of ihe subject and predicate is not merely asserted, but the subject is declared to be aname given to all those magnitudes which have the properties laid down in the predicate, and to no others. Thus a square is a four-sided figure having equal sides and one right angle, that is, let every four-sided figure having &c., be called a square, and let no other figure be called by that name, whence the truth of the converse is evident. Also many of the facts proved in geometry are conversely true. Thus all equilateral triangles are equiangular, from which it is proved that ald equiangular triangles are equilateral. Of the first species of assertion, the universal affirmative, the converse is not necessarily true. Thus ‘every point in figure A is a point of B,’ does not imply that ‘ every point of B is a point of A,’ although this may be the case, and is, if the two figures coincide entirely. The second species, the parti- cular affirmative, is conversely true, since if some points of A are points of B, some points of B are also points of A. The first species of assertion is con- versely true, if the converse be made to take the form of the second species: thus from ‘ allright angles are equal,’ it may 69 be inferred that ‘ some equal magnitudes are right angles.’ The third species, the universal negative, is conversely true, since if ‘no point of B isa point of A,’ it may be inferred that ‘no point of is a point of B. The fourth species, the particular negative, is not neces sarily conversely true. From ‘ some points of A are not points of B,’ or A is not entirely contained within B, we can infer nothing as to whether B is or is not entirely contained in A. It is plain that the converse of a proposition is not necessarily true, if it says more either of the subject or predicate than was said before. Now ‘every equi- lateral triangle is equiangular,’ does not speak of all equiangular triangles, but asserts that among all possible equiangular triangles are to be found add the equilateral ones. There may then, for any thing to the contrary to be dis- covered in our assertion, be classes of equiangular triangles not included under this assertion, of which we can therefore say nothing. But in saying ‘ no right angles are unequal,’ that which we ex- clude, we exclude from all unequal angles, and therefore ‘no unequal angles are right angles’ is not more general than the first. The various assertions brought forward in a geometrical demonstration must be derived in one of the following ways. I. From definition. This is merely sub- stituting, instead of a description, the name which it has been agreed to give to whatever bears that description. No definition ought to be introduced until it is certain that the thing defined is really possible. Thus though parallel lines are defined to be ‘lines which are in the same plane, and which being ever so far produced never meet,’ the mere agreement to call such lines, should they exist, by the name of parallels, is no sufficient ground to assume that they do exist. The definition is therefore inadmissible until it is really shewn that there are such things as lines which being in the same plane never meet. Again, before applying the name, care must be taken that all the circumstances connected with the definition have been attended to. Thus, though in plane geo- metry, where all lines are in one plane, it is sufficient that two lines would never meet though ever so far produced, to call them parallel, yet in solid geometry the first circumstance must be attended to, and it must be shewn that lines are in the same plane before the name can be 70 applied. Someof the axioms come so near to definitions in their nature, that their place may be considered as doubt- ful. Such are, ‘the whole is greater than its part,’ and ‘magnitudes which entirely coincide are equal to one ano- ther.’ II. From hypothesis. In the statement of every proposition, certain connexions are supposed to exist from which it is asserted that certain consequences will follow. Thus ‘in an isosceles triangle the angles at the base are equal,’ or, ‘if a triangle be isosceles the angles at the base will be equal.’ Here the hypo- thesis or supposition is that the triangle has two equal sides, the consequence as- serted is that the angles at the base or third side will be equal. The conse- quence being only asserted to be true when the angle is isosceles, such a tri- angle is supposed to be taken as the basis of the reasonings, and the condition that its two sides are equal, when introduced in the proof, is said to be introduced by hypothesis, In order to establish the result it may be necessary to draw other lines, &c., which are not mentioned in the first hy- pothesis. These, when introduced, form what is called the construction. There is another species of hypothesis much in use, principally when it is required to deduce the converse of a theorem from the theorem itself. In- stead of proving the consequence di- rectly, the contradictory of the conse- quence is assumed to hold good, and if from this new hypothesis, supposed to exist together with the old one, any evidently absurd result can be derived, such as that the whole is greater than its part, this shews that the two hypotheses are not consistent, and that if the first be true, the second cannot be so. But if the second be not true, its contradictory is true, which is what was required to be proved, III. From the evidence of the asser tions themselves, The propositions thus introduced without proof are only such as are in their nature too simple to admit of it. They are called axioms. But it is necessary to observe, that the claim of an assertion to be called an axiom does not depend only on its being self-evident. Were this the case many propositions which are always proved might be assumed ; for example, that two sides of a triangle are greater than the third, or that a straight line is the shortest distance between two points, STUDY OF Tn addition to being self-evident, it must be incapable of proof by any other means, and it is one of the objects of geometry to reduce the demonstrations | to the least possible number of axioms, | There are only two axioms which are distinctly geometrical in their nature, | viz., ‘two straight lines cannot enclose a space,’ and ‘ through each point outside’ a line, not more than one parallel to. that line can be drawn.’ All the rest of the propositions commonly given as axioms are either arithmetical in their nature ; such as ‘the whole is greater than its part,’ ‘the doubles of equals are equals,’ &c.; or mere definitions, such as ‘ magnitudes which entirely coincide are equal;’ or theorems ad- mitting of proof, such as ‘all right angles are equal.’ There is however one more species of self-evident proposition, the postulate or self-evident problem, such as the possibility of drawing a right line, &e. IV. From proof already given. What has been proved once may be always taken for granted afterwards. It is evident that this is merely for the sake of brevity, since it would be possi- ble to begin from the axioms and pro- ceed direct to the proof of any one pro- position, however far removed from them ; and this is an exercise which we recommend to the student. Thus much for the legitimate use of any single as- sertion or proposition. We proceed to the manner of deducing a third propo- sition from two others. It is evident that no assertion can be the direct and necessary consequence of two others, unless those two contain something in common, or which is spoken of in both. In many, nay most cases of ordinary conversation and writ- ing, we leave out one of the assertions, which is, usually speaking, very evident, and make the other assertion followed by the consequence of both. Thus, ‘ geo- metry is useful, and therefore ought to be studied,’ contains not only what is expressed, but also the following, ‘that which is useful ought to be studied ;’ for were this not admitted, the former assertion would not be necessarily true. This may be written thus,— Every thing useful is what ought to be studied. ; Geometry is useful, therefore geome- try is what ought to be studied. This, in its present state, is called ‘a syllogism, and may be compared with the following, from which it only differs MATHEMATICS. _ in the things spoken of, and not in the manner in which they are spoken of, Every point of the circle is a point of the triangle. The point B is a point of’the circle. Therefore the point B is a point of the triangle. Here a connexion is esta- plished between the point B and the points of the triangle (viz. that the first is one of the second) by comparing them with the points of the circle; that which is asserted of every point of the circle in the first can be asserted of the point B, because from the second B is one of these points. Again, in the former ar- -gument, whatever is asserted of every thing useful is true of geometry, because geometry is useful. The common term of the two propo- sitions is called the middle term, while the predicate and subject of the conclu- sion are called the major and minor terms, respectively. The two first asser- tions are called the major and minor premises, and the last the conclusion. Suppose now the two premises and con- clusion of the: syllogism just quoted to be varied in every possible way from _ affirmative to negative, from universal to particular, and vice versa, where the number of changes will be 4 x4 x 4, or 64 (called moods); since each pro- position may receive four different forms, The sum of the squares of the lines a and 6 and the square of the line c a? + 6) ana e fate a? -- 62 Therefore Bide e Here the term square in the major premiss has its geometrical, and in the minor its algebraical sense, being in the first a geometrical figure, and in the second an arithmetical operation. The ‘term of comparison is not therefore the same in both, and the conclusion does not therefore follow from the pre- mises. The same error is committed if all that can be contained under the middle term be not spoken of either in the ma- jor or minor premiss. For if each pre- miss only mentions a part of the middle term, these parts may be different, and the term of comparison really different in the two, though passing under the same name in both. Thus, All the triangle is in the circle, All the square is in the circle, proves nothing, since the square may, 71 and each form of one may be com- pounded with any of the other two. And these may be still further varied, if instead of the middle term being the subject of the first, and the predicate of the second, this order be reversed, or if the middle term be the subject of both, or the predicate of both, which will give four different figures, as they are called, to each of the sixty-four moods above mentioned. But of these very few are correct deductions, and without enter- ing into every case we will state some general rules, being the methods which common reason would take to ascertain the truth or falsehood of any one of them, collected and generalised *. I. The middle term must be the same in both premises, by what has just been observed; since in the comparison of two things with one and the same third thing, in order to ascertain their con- nexion or discrepancy, consists the whole of reasoning. Thus, the deduction with- out further process of the equation a? + b2 = c? from the proposition, which proves that the sum of the squares de- scribed on the sides of a right-angled triangle is equal to the square on its hy- pothenuse, a, b, and ¢ being the number of linear units in the sides and the hypo- thenuse, is incorrect, since syllogistically stated the argument would stand thus ;—~ _. fequal j ae (ea eas Re sum of the squares of a and 6, and the square of c. equal j sis {ea ties consistently with these conditions, be either wholly, partly, or not at all, con- tained in the triangle. In fact, as we have before shewn, each of these asser- tions speaks of a part of the circle only. The following is of the same kind. Some of the triangle is in the circle. Some of the circle is not in the square, oF II. If both premises are negative, no conclusion can be drawn. For it can evi- dently be no proof either of agreement or disagreement that two things both disagree with a third. Thus the follow- ing is inconclusive,— None of the circle is in the triangle. None of the square is in the circle. III. If both premises are particular, ers Ped Sh ee * Whately’s Logic, page 76, third edition. A work which should be read by all mathematical students, 72 no conclusion can be drawn, as will ap- pear from every instance that can be taken: thus,— Some of the circle is in the triangle, Some of the square is not in the cir- cle, proves nothing. IV. In forming a conclusion, where a conclusion can be formed, nothing must be asserted more generally in the con- clusion than in the premises. Thus, if from the following, All the triangle is in the circle, All the cirele is in the square, we would draw a conclusion in which the square should be the subject, since the whole square is not mentioned in the minor premiss, but only part of it, the conclusion must be, Part of the square is in the triangle. V. If either of the premises be nega- tive, the conclusion must be negative. For as both premises cannot be nega- tive, there is asserted in one premiss an agreement between the term of the con- clusion and the middle term, and in the other premiss a disagreement between the other term of the conclusion, and the same middle term. From these nothing can be inferred but a disagreement or negative conclusion. Thus, from None of the circle is in the triangle, All the circle is in the square, can only be inferred, Some of the square is zo¢ in the tri- angle. VI. If either premiss be particular, the conclusion must be particular, For example, from None of the circle is in the triangle, Some of the circle is in the square, we deduce, Some of the square is not in the tri- angle. If the student now applies these rules, he will find that of the sixty-four moods eleven only are admissible, in any case; and in applying these eleven moods to the different figures he will also find that some of them are not admissible in every figure, and some not necessary, on account of the conclusion, though true, not being as general as from the premises it might be. This he may do either by reasoning or by actual inspec- tion of the figures, drawn and arranged according to the premises. The ad- missible moods are nineteen in number, and are as follows, where A at the beginning of a proposition signifies that it is a universal aftirmative, E a univer- sal negative, Ia particular affirmative, O a particular negative. STUDY OF Figure I, The middle term is the sub- ject of the major, and the predicate of the minor premiss. 1,*A_ All the () isin the A A All the [7 is in the C) + ery Aliathie Cis in the A 2. E Noneofthe() is in the A A All the [J is in the C) “. FE Noneofthe is in the A 3. A All the C) is in the A I Some of the is in the CE “. I Some ofthe is in the A 4. E Noneofthe() is in the A I Some of the is in the C) “. O SomeoftheWis not in A Figure IT. The middle term is the predicate of both premises. 1. E Noneofthe A is in the A All the [1 is in the esa None of the is in the A 2: All the A is in the None of the & is in the ¢ None of the is in the A None of the A is in the Some of ther is in the Some of the Mis not in A All the A is in the Some of the is not in Some of the [is not in A Figure III. The middle term is the subject of both premises. 1, All the isin the A All the is in the Some ofthe is in the A Some of the C) is in the A All the () is in the c9 Some ofthe is in the A All the is in the A Some of the(_) is in the C9 Some of the is in the A None of the() is in the A All the is in the Some of the isnot in A is notin A Some of the All the is in the Cy Some of ther is not in A None of the) is in the A Some of the() is in the es Some of the is not in A Figure IV. The middle term is the predicate of the major, and the subject of the minor premiss. ko OHE OPO OP HED HDHD ae This, and 3, are the most simple of all the com. binations, and the most frequently used, especially in geometry: MATHEMATICS, All the A is in the C) Allthe (Q)isin thet Some of the is in the A All the A is in the C) None of the) is in the & None ofthe is in the A Some ofthe A is in the) Allthe ()is in them Some of the is in the A None of the A is in the C) Allthe” () isin them SomeoftheCcis notin A None ofthe A is in the () Someofthe() is in the Some of the is not in A OnH OPH HRPH HE> HPP We may observe that it is sometimes possible to condense two or more syllo- gisms into one argument, thus: Every A is B (1), Every B is C (2), Every C is D (3), Every D is E (4), Therefore Every A is E (5), is equivalent to three distinct syllogisms of the form Figure I.; these syllogisms at length being 1, 2,a; a, 3,0; 6,4, 5. The student, when he has well con- sidered each of these, and satisfied him- self, first by the rules, and afterwards by inspection, that each of them is legi- timate; and also that all other moods, not contained in the above, are not al- lowable, or at least do not give the most general conclusion, should form for himself examples of each case, for instance of Fig. IIT. 3. The axioms constitute part of the basis of geometry. Some of the axioms are grounded on the evidence of the senses. ~.Some evidence derived from the senses is part of the basis of geometry. He should also exercise himself in the first principles of reasoning by reducing arguments as found in books to the syl- logistic form. Any controversial or argu- mentative work will furnish him with a sufficient number of instances. Inductive reasoning is that in whicha universal proposition is proved by pro- ving separately every one ofits particular cases. As where, for example, a figure, ABCD, is proved to be a rectangle by proving each ofits angles separately to be a right angle, or proving all the premises of the following, from which the conclusion follows necessarily, 73 The angles at A, B, C, and D are all the angles of the igure ABC D. A is a right angle, B is aright angle, | C is a right angle, Dis aright angle, Therefore all the angles of the figure ABCD areright angles. This may be considered as one syllo- gism of which the minor premiss is, A,B, C,and D areright angles, where each part is to be separately proved. Reasoning a fortiorz, is that contained in Fig. I. 1. in a different form, thus: A is greater than B, B is greater than C ; a fortiort Ais greater than C; which may be also stated as follows: The whole of B is contained in A, The whole of C is contained in B, Therefore C is contained in A. The premises of the second do not ne- cessarily imply as much as those of the first ; the complete reduction we leave to the student. The elements of geometry present a collection of such reasonings as we have just described, though in a more con- densed form. It is true that, for the convenience of the learner, it is broken up into distinct propositions, as a jour- ney is divided into stages; but neverthe- less, from the very commencement, there is nothing which is not ofthe nature just described. We present the following as a specimen of a geometrical propo- sition reduced nearly to a syllogistic form. To avoid multiplying petty syllo- gisms, we have omitted some few which the student can easily supply. (Treatise on Geometry, page 21.) Hypornssis. AB C is aright-angled triangle the right angle being at A. CONSEQUENCE. The squares on AB and AC are together equal to the square on BC, Construction. Upon BC and BA describe squares, produce DB to meet EF, produced, if necessary, in G, and draw HK parallel to BD. 74 Demonstration. * T. Conterminous sides of a square are at right angles to one another. (Defi- nition.) E B and B A are conterminous sides of a square. (Construction.) .. E B andB A are at right angles. II. A similar syllogismto prove that DB and B C are at right angles, and another to prove that G B and BC are at right angles. Ill. Two right lines drawn perpen- dicular to two other right lines make the same angle as those others (already proved); E Band BG and AB and BO are two right lines &ce., (1. II). . The angle E B Gis equal to ABC. IV. All sides of a square are equal. (Definition.) AB and B E are sides of a square. (Construction.) .-. A B and B E are equal. V. Allright angles are equal. ready proved.) BE Gand BAC are right angles. (Hypothesis and construction.) .. BEG and BA C are equal angles. VI. Two triangles having angles of one equal to two angles of the other, and the interjacent sides equal, are equal in all respects. (Proved.) BEG and B A C are two triangles having B EG and E B G respectively equal toB AC and A BC and the sides EB and B A equal. (III. 1V. V.) .. The triangles BEG, B A C are equal in all respects. VII. B Gis equaltoBC. (VI) BC is equalto BD. (Proved as IV.) ». BGis equal to BD. f VIII. A four-sided figure whose op- posite sides are parallel is a parallelo- (Al- STUDY OF gram. (Definition) BGH A‘and_ BPKD are four-sided figures &c._ (Construction.) | “.BGHAandBP KD are parallelo- | grams. | IX. Parallelograms upon the same base and between the same parallels are equal. (Proved.) EBAF and B GHA, are parallelograms &c. (Construction.) ~. EBA F and BGHA are equal. X. Parallelograms on equal bases and between the same parallels, are equal. (Proved.) BGHAand BDKP are parallelo- grams &e. (Construction.) . -. BGHA and BD K P are equal. XI. EBAFisequalto BOGHA. (1X) BGHA is equal to BD K P. (X,) «. EBA F (that is the square on A B) isequal to BD K P. XII. A similar argument from the commencement to prove that the square on A C is equal to the rectangle C PK. XIII. The rectangles BK and CK are together equal to the square on AB, (Self-evident from the construction.) The squares on BA and AC are to- gether equal to the rectangles B K and CK. (Self-evident from XI and XII.) _ . The squares on BA and AC are together equal to the square on BA. Such is an outline of the process, every step of which the student must pass through before he has understood the demonstration. Many of these steps are not contained in the book, because the most ordinary intelligence is suffi; cient to suggest them, but the least is as necessary to the process as the greatest, Instead of writing the propositions at this length, the student is recommended to adopt the plan which we now lay before him. AB C isa triangle, right angled at A. a On BA describe a square BAF E. On BC describe a square. Produce BD to meet EF, produced if neces- sary in G. Hyp. 1 Constr. - 3 a 4 5 b Demonst. 6 2, Def. 7 3 8 6, 75°C 90 Ay hed - 10 2 11y 8,83 10; 12 1l, 1Gan cog, ae 14. 5935DVel: 6x 18,25 16 13, 14, g 17. —=—«.15, 16 Through A draw H AK parallel to BD. EBA is aright angle. GB Cis a right angle. Z EBG is equalto Z ABC. Z BEG is equalto Z BAC. EB is equal to AB. 0 The triangles BE G and A B C are equal. 3 BG is equal to BD. f. A HGB isa parallelogram. f. BPDK isa parallelogram. AHGB and ABEF are equal. AHGB and BPD K are equal. BPD K and the square on A B are equal. MATHEMATICS. 18 { By similar reasoning 19 17, 18 75 } C PK and the square on C A are equal. The square on B C is equal to the square on BA and AC. a, b Here refer to the necessary problems. ¢ If two lines be drawn at right angles to two others, the an- gles made by the first and second pair are equal. d Allright angles are equal. e ‘Two triangles which have two angles of one equal to two angles of the others, and the interjacent side equal, are equal in all respects. Parallelograms on the same or equal bases, and between the same parallels, are equal. The explanation of this is as follows: the whole proposition is divided into distinct assertions, which are placed in Separate consecutive paragraphs, which paragraphs are numbered in the first column on the left ; in the second column on the left we state the reasons for each paragraph, either by referring to the preceding paragraphs from which they follow, or the preceding propositions in which they have been proved. In the latter case a letter is placed in the column, and at the end, the enunciation of the proposition there used is written opposite to the letter. By this method, the proposition is much shortened, its more prominent parts are brought im- mediately under notice, and the beginner, if he recollect the preceding propositions perfectly well, is not troubled by the re- petition of prolix enunciations, while in the contrary case he has them at hand for reference. In all that has been said, we have taken instances only of direct reasoning, that is, where the required result is im- mediately obtained without any reference to what might have happened if the result to be proved had not been true. But there are many propositions in which the only possible result is one of two things which cannot be true at the same _ time, and it is more easy to shew that - one is vot the truth, than that the other as. This is called indirect reasoning ; not that it is less satisfactory than the first species, but because, as its name imports, the method does not appear so direct and natural, There are two pro- positions of which it is required to shew that whenever the first istrue the second is true ; that is, the first being the hypo- thesis the second is a necessary conclu- sion from it, whence the hypothesis in question, and any thing contradictory to, or inconsistent with the conclusion can- not exist together. In indirect reasoning, we suppose that, the original hypothesis existing and being true, something incon. sistent with or contradictory to the conclusion is true also. If from com- bining the consequences of these two suppositions, something evidently erro- neous or absurd is deduced, it is plain that there is something wrong’in the assumptions. Now care is taken that the only doubtful point shall be the one just alluded to, namely, the supposition that one proposition and the contra- dictory of the other are true together. This then is incorrect, that is, the first proposition cannot exist with any thing contradictory to the second, or the second must exist wherever the first exists, since if any proposition be not true its contradictory must be true, and vice versa. This is rather embarrassing to the beginner, who finds that he is required to admit, for argument’s sake, a proposition which the argument itself goes to destroy. But the difficulty would be materially lessened, if instead of assuming the contradictory of the second proposition positively, it were hypothetically stated, and the conse- quences of it asserted with the verb ‘would be,’ instead of ‘is. For example: suppose it to be known that if A is B, then C must be D, and it is required to shew indirectly that when C is not D, A is not B. This put into the form in which such a proposition would appear in most elementary works, is as follows. It being granted that if A is B, Cis D, it is required to shew that when C is not D, A is not B. If possible, let C be not D, and let Abe B. Then by what is granted, since A is B, Cis D; but by hy- pothesis C is not D, therefore both C 7s and 7s zot D, which is absurd; that is, it is absurd to suppose that C.zs not D and A zs B, consequently when C is not D, A isnot B. The following, which is exactly the same thing, is plainer in its language. Let C be not D. Then if A were B, C would be D by the propo- sition granted, But by hypothesis C is 16. not D,&e. This sort of indirect rea- soning frequently goes by the name of reductio ad absurdum. In all that has gone before we may perceive that the validity of an argument depends upon two distinct consider- ations,—1l, the truth of the relations as- sumed, or represented to have been proved before; 2, the manner in which these facts are combined so as to produce new relations ; in which last the 7eason- ing properly consists. If either of these be incorrect in any single point, the result is certainly false ; if both be incorrect, or if one or both be incorrect in more points than one, the result, though not at all to be depended on, is not certainly false, since it may happen and has happened, that of two false reasonings or facts, or the two combined, one has reversed the effect of the other and the whole result has been true; but this could only have been ascer- tained after the correction of the erro- neous fact or reasoning. The same thing holds good in every species of rea soning, and it must be observed, that how- ever different geometrical argument may be in form from that which we employ daily, it is not different in reality. Weare accustomed to talk of mathematical rea- soning as above all other, in point of accuracy and soundness. This, if by the term reasoning we mean the com- paring together of different ideas and producing other ideas from the com- parison, is not correct, for in this view mathematical reasonings and all other reasonings correspond exactly. For the real difference between mathematics and other studies in this respect we refer the student to the first chapter of this treatise. In what then, may it be asked, does the real advantage of mathematical study consist ? We repeat again, in the actual certainty which we possess of the truth of the facts on which the whole is based, and the possibility of verifying every result by actual measurement, and not in any superiority which the method of reasoning possesses, since there is but one method ofreasoning. To pursue the illustration with which we opened this work (page the first), suppose this point to be raised, was the slaughter of Cesar justifiable or not? The actors in that deed justified themselves by saying, that a tyrant and usurper, who meditated the destruction of his coun- try’s liberty, made it the duty of every citizen to put him to death, and that STUDY OF Ceesar was a tyrant and usurper, &e. Their reasoning was perfectly correct, though proceeding on premises then extensively, and now universally denied. The first premiss, though correctly used in this reasoning, is now asserted to be false, on the ground that itis the duty of every citizen to do nothing which would, were the practice universal, militate against the general happiness ; that were each individual to act upon his own judgment, instead of leaving offenders to the law, the result would be anarchy and complete destruction of civilization, &e. Now in these reasonings and all others, with the exception of those which occur in mathematics, it must be ob- served that there are no premises so certain, as never to have been denied, no first principles to which the same degree of evidenceis attached as to the following, that ‘ no two straight lines can enclose a space.” In mathematics, therefore, we reason on certainties, on notions to which the name of innate can be applied, if it can be applied to any whatever. Some, on observing that we dignify such simple consequences by the name of reasoning, may be loth to think that this is the process to which they used to attach such ideas of difficulty. There may, perhaps, be many who imagine that reasoning is for the mathematician, the logician &e., and who, like the Bour- geois Gentilhomme, may be surprised on being told, that, well or ill, they have been reasoning all their lives. And yet such is the fact ; the commonest actions © of our lives are directed by processes exactly identical with those which enable us to pass from one proposition of geo- metry to another. A porter, for example, who being directed to carry a parcel from the City to a street which he has never heard of, and who on inquiry, finding it is in the Borough, concludes that he must cross the water to get atit, has performed an act of reasoning, differing nothing in kind from those by a series of which, did he know the previous propositions, he might be convinced that the square of the hypothenuse of a right-angled triangle, is equal to the sum of the squares of the sides. CHAPTER XV. On Axioms. GromeTRY, then, is the application of strict logic to those properties of space and figure which are self-evident, and which therefore cannot be disputed. MATHEMATICS. But the rigour of this science is carried one step further; for no property, how- ever evident it may be, is allowed to pass without demonstration, if that can be given. The question is therefore to de- ’ monstrate all geometrical truths with the smallest possible number of as- sumptions. These assumptions are called axioms, and for an axiom it is requi- site, —1, that it should be self-evident; 2, that it should be incapable of being proved from the other axioms. In fulfilling these conditions, the number of axioms which are really geometrical, that is, which have not equal reference to Arith- metic, is reduced to two, viz., two straight lines cannot enclose a space, and through a given point not more than one pa- rallel can be drawn to a given straight line. The first of these has never been considered as open to any objection; it has always passed as perfectly self-evi- dent. It is on this account made the proposition on which are grounded all reasonings relative to the straight line, since the definition of a straight line is too vague to afford any information, But the second, viz., that through a given point not more than one parallel can be drawn to a given straight line, has always been considered as an assumption not self-evident in itself, and has therefore been called the defect and disgrace of geo- metry. We proceed to place it on what we conceive to be the proper footing. By taking for granted the arithmetical axioms only, with the first of those just alluded te, the following propositions may be strictly shewn. I. One perpendicular and only one ean be let fall from any point A to agiven line CD. Let this be A B. II. If equal distances B C and B D be faken on both sides of B, A C and AD are equal, as also the angles B A C and BAD. III, Whatever may be the length of BC and BD, the angles BAC and BAD are each less than a right angle. IV. Through A, a line may be drawn parallel to CD (that is, by definition, never meeting C D, though the two be ever so far produced), by drawing any line AD and making the angle DAE 417 equal to the angle ADB, which it is before shewn how to do. From proposition(IV) we should at first see noreason against there being as many parallels to C D, to be drawn through A, as there are different ways of taking A D, since the direction for drawing a parallel to C Dis, ‘take any line AD cutting C D and make the angle D AE equal to ADB. But this our senses immedi- ately assure us is impossible. It appears alsoa proposition to which no degree of doubt can attach, that if the straight line AB, produced indefi- nitely both ways, set out from the po- sition AB and revolve round the point A, moving first towards AE; then the point of intersection D will first be on one side of B and afterwards on the other, and there will be one position where there is no point of intersection either on one side or the other, and one such position only. ‘This is in reality the assumption of Euclid ; for having proved that AE and BF are parallel -when the angles BDA and DAE are equal, or, which is the same thing, when EAD andADF are together equal to two right angles, he further assumes that they will be parallel in no other case, that is, that they will meet when the angles EAD and AD F are together greater or less than two right angles ; which is really only assuming that the parallel which he has found, is the only one which can be drawn. The remaining part of his axiom, namely, that the lines AE and DE, if they meet at all, will meet upon that side of D A on which the angles are less than two right angles, is not an assumption but a consequence of his proposition, which shews that any two angles of a triangle are together less than two right angles, and which is established before any mention is made of parallels. It has been found by the experience of two thousand years, that some assumption of this sort is indis- pensable. Every species of effort has been made to avoid or elude the diffi- culty, but hitherto without success, as some assumption has always been in- volved, at least equal, and in most cases superior in difficulty to the one already made by Euclid. For example, it has been proposed to define parallel lines as those which are equidistant from one another at every point. In this case, before the name parallel can be allowed to belong to any thing, it must be proved that there are lines such that a per- pendicular to one is always perpen 78 dicular to the other, and that the parts of these perpendiculars intercepted be- tween the two are always equal. A proof of this has never been given without the ‘previous assumption of something equivalent to the axiom of Euclid. Of this last, indeed, a proof has been given, but involving considerations not usually admitted into geometry, though it is more than probable that had the same come down to us, sanc- tioned by the name of Euclid, it would have been received without diffi- culty. The Greek geometer confines his notion of equal magnitudes to those which have boundaries. Suppose this notion of equality extended to all such spaces as can be made to coincide entirely in all their extent, whatever that extent may be; for example, the unbounded spaces contained between two equal angles whose sides are produced without end, which by the definition of equal angles might be made to coincide en- tirely by laying the sides of one angle upon those of the other. In the same sense we may say, that, one angle being double of another, the spaces contained by the sides of the first is double that contained by the sides of the second, and so on. Now suppose two lines Oa and Od, making any angle with one another, and produced ad infinitum *. On Oa take off the equal spaces OP, PQ, QR, &c., ad infinitum, and draw the lines Pp, Q gq, Rr, &c., so that the angles OP p, OQ q &e., shall be equal to one another, each being such as with 60 P will make two right angles. Then Ob, Pp, Qq, &c., are parallel to one another, and the infinite spaces 5O Pp, pPQq, gQRr, &e., can be made to coincide, and are equal. Also no finite number whatever of these spaces will fill up the infinite space 5 Oa, since OP, PQ, &c. may be continued ad infinitum upon the line Oa. Let there be any * Every linein this figure must be produced ad infinitum, from that extremity at which the small letter is placed, STUDY OF line O#, such that the” angles OP and p PO are together less than two right angles, that is, less than 6O Pand p PO; whence ¢O Pis less than 6 O P and ¢O falls between 00 andaO. Take the angles Ov, vOw, wOa, &c., each equal to 602, and continue this until the last line Oz falls beneath Oa, so that the angle 602 is greater than bOa. That this is possible needs no proof, since it is manifest that any angle being continually added to itself the sum will in time exceed any other given angle ; again, the infinite spaces DOZ, Ov, &e., are all equal. Now on comparing the spaces bO¢ and bOPp, we see that a — certain number of the first is more than equal to the space Oa, while no num- ber whatever of the second is so great. We conclude, therefore, that the space 60¢ is greater than 50Pp, which cannot be unless the line O¢ cuts Pp © at last; for if Of did never cut Py, the space bO#¢ would evidently be less than © 60 Pp», as the first would then fall en- tirely within the second. Therefore two lines which make with a third angles © together less than two right angles will meet if sufficiently produced. This demonstration involves the con- sideration of a new species of magnitude, namely, the whole space contained by the sides of an angle produced without — limit. This space is unbounded, and is — greater than any number whatever of finite spaces, of square feet, for exam- ple. No comparison, therefore, as to magnitude, can be instituted between if and any finite space whatever, but that affords no reason against comparing — this magnitude with others of the same kind. Any thing may become the subject of © mathematical reasoning, which can be increased or diminished by other things of the same kind; this is, in fact, the definition given of the term magnitude ; and geometrical reasoning, in all other cases at least, can be applied as soon as a criterion of equality is discovered. Thus the angle, to beginners, is a per- fectly new species of magnitude, and one of whose measure they have no conception whatever ; they see, however, that it is capable of increase or diminu- tion, and also that two of the kind ean be equal, and how to discover whether this is so or not, and nothing more is necessary for them. All that can be said of the introduction of the angle in geometry holds with some, to us it ap- pears an equal force, with regard to MATHEMATICS these unlimited spaces; the two are ' very closely connected, so much so, that _ the term angle might even be defined as ‘the unlimited space contained by two right lines,’ without alteration in the truth of any theorem in which the word angle is found. But this is a point which cannot be made very clear to the beginner. The real difficulties of geometry be- gin with the theory of proportion, to _ which we now proceed. The points of | discussion which we have _ hitherto raised, are not such as to embarrass the elementary student, however much they may perplex the metaphysical in- quirer into first principles. The theory _ to which we are coming abounds in difficulties of both classes. CHAPTER XVI. On Proportion: In the first elements of geometry, two lines, or two surfaces, are mentioned in no other relation to one another, than that of equality or non-equality. Nothing but the simple fact is announced, that one magnitude is equal to, greater than, or less than another, except occasionally when the sum of two equal magnitudes is said to be double of one of them. Thus in proving that two sides of a tri- angle are together greater than the third, the fact that they are greater is the essence of the proposition; no mea- sure is given of the excess, nor does anything follow from the theorem as to _ whether it is, or may be, small or great. ’ : We now come to the doctrine of pro- portion, in which geometrical magnitude is considered in a new light. The sub- ject has some difficulties, which have been materially augmented by the al- most universal use, in this country at least, of the theory laid down in the fifth book of Euclid. Considered as a com- plete conquest over a great and ac- knowledged difficulty of principle, this book of Euclid well deserves the immor- tality of which its existence, at the pre- sent moment, is the guarantee; nay, had the speculations of the mathema- tician been wholly confined to geome- trical magnitude, it might be a question whether any other notions would be necessary. But when we come to apply arithmetic to geometry, it is necessary to examine well the primary connexion between the two; and here difficulties arise, not in comprehending that con- hexion, so much as in joining the two 79 sciences by a chain of demonstration as strong as that by which the propositions of geometry are bound together, and as little open to cayil and disputation. The student is aware that before pro- nouncing upon the connexion of two lines with one another, it is necessary to measure them, that is, to refer them to some third line, and to observe what number of times the third is contained in the other two. Whether the two first are equal or not is readily ascertained by the use of the compasses, on princi- ples laid down with the utmost strictness in Euclid, and other elementary works. But this step is not sufficient; to say that two lines are not equal, determines nothing. There are an infinite number of ways in which one line may be greater or less than a given line, though there is only one in which the other can be equal to the given one. We proceed to shew how, from the common notion of measuring a line, the more strict geo- metrical method is derived. To measure the line AB apply to it ano- ther line (the edge of a ruler), which is divided into equal parts (as inches), each of which parts is again subdivided into ten equal parts, as in the figure. This division is made to take place in prac- tice until the last subdivision gives a part so small, that any thing less may be neglected as inconsiderable. Thus a carpenter's rule is divided into tenths or eighths of inches only, while in the tube of a barometer a process must be em- ployed which will mark a much less difference. In talking of accurate mea- surement, therefore, any where but in geometry, or algebra, we only mean ac- curate as far as the senses are con- cerned, and as far as is necessary for the object in view. The ruler in the figure shews that the line AB contains more than two and less than three inches; and closer inspection shews that the excess above two inches is more than six-tenths of an inch, and less than seven. Here, in practice, the process stops; for, as the subdivision of the ruler was carried only to tenths of 80 inches,. becatise a tenth of an inch is a quantity which may be neglected in or- dinary cases, we may call the line two inches and six-tenths, by doing which the error committed is less than one- tenth of aninch. In this way lines may be compared together with a common degree of correctness; but this is not enough for the geometer. His notions of accuracy are not confined to tenths or hundredths, or hundred-millionth parts of any line, however small it may be at first. The reason is obvious; for although to suit the eye of the gene- rality of readers, figures are drawn, in which the least line is usually more than an inch, yet his theorems are as- serted to remain true, even though the dimensions of the figure are so far di- minished as to make the whole imper- ceptible in the strongest microscope. Many theorems are obvious upon look- ing at a moderately-sized figure; but the reasoning must be such as to con- vince the mind of their truth when, from excessive increase or diminution of the scale, the figures themselves have past the boundary even of imagination. The next step in the process of measurement is as follows, and will lead us to the great and peculiar difficulty of the sub- yect. The inch, the foot, and the other lengths, by which we compare lines with one another, are perfectly arbitrary. There is no reason for their being what they are, unless we adopt the commonly received notion that our inch is derived from our Saxon ancestors, who ob- served that a barley-corn is always of the same length, or nearly so, and placed three of them together as a common standard of measure, which they called an inch. Any line whatever may be chosen as the standard of measure, and it is evident that when two or more lines are under consideration, exact compa- risons of their lengths can only be ob- tained from a line which is contained an exact number of times in them all. For even exact fractional measures are reduced to the same denominator, in order to compare their magnitudes. Thus, twolines which contain -2, and 3 of a foot, are better compared by observing that =2, and 3 being 14 and 33, the given lines contain one 77th part of a foot 14 and 33 times respectively. Any line which is contained an exact number of timesin another is called in geometry a measure of it, and a common measure STUDY OF of two or more lines is that which is cori« ! tiined an exact number of times in each. Again, a line which is measured by another is called a multiple of it, as in arithmetic. The same definition, mutatis mutandis, applies to surfaces, solids, and all other magnitudes; and though, in our suc- ceeding remarks, we use lines as an illustration, it must be recollected that the reasoning applies equally to every | magnitude which can be made the sub- | ject of calculation. In order that two quantities may ad- mit of comparison as to magnitude, they | must be of the same sort; if one isa ° line, the other must be a line also. Suppose two lines A and B each of | which is measured by the line C; the first containing it five times and the second six. These lines A and B, which contain the same line C five and six times respectively, are said to have to one another the ratio of five to six, or to bein the proportion of five to six. If then we denote the first by A*, and the second by B, and the common measure by C, we have A=5C, or6A= 30C, B=6C, 5B = 30C, whence 6A =5B, or6A— 5B=0. Generally, when mA — 2B = 0, the lines, or whatever they are, represented by A and B, are said to be in the pro- | portion of m to m, or to have the ratio of n to m. ? Let there be two other magnitudes P and Q, of the same kind with one another, either differing from the first in kind or not; thus A and B may be lines, and P and Q surfaces, &c., and let them contain a common measure R, just as A and B contain C, viz.: Let P- contain R five times, and let Q contain R six times, we have by the same reasoning 6P—5Q =0, and P and Q, being also in the ratio of * The student must distinctly understand that the common meaning of algebraical terms is departed from in this chapter, wherever the letters are large instead of small. Forexample, A, instead of mean- ing the number of units of some sort or other contain- ed in the line A, stands for the line A itself, and mA (the smali letters throughout meaning whole num- bers) stands for the line made by taking A, m times. Thus such expressions as mA + b, mA — 7B, &c., are the only ones admissible. AB, =: A?, &c., are unmeaning, while — is the line which is contained Mm 7 m times in A, or the mth part of A. The capital letters throughout stand for concrete quantities, not for their representations in abstract numbers, MATHEMATICS. five to six, as wellas A and B, are said to be proportional to A and B, which is de- noted thus, ee tes Poe CD, by which at present all we mean is this, that there are some two whole numbers mand 7 such that, at the same time mA —nB = 0, mP —- nQ=0, Nothing more than this would be ne- cessary for the formation of a complete theory of proportion, if the common measure, which we have supposed to exist in the definition, did always really exist. We have however no right to as- sume, that two lines A and B, whatever may be their lengths, both contain some other line an exact number of times. We can moreover produce a direct instance in which two lines have no common measure whatever, in the fol- - lowing manner. A aot Let A B C bean isosceles right-angled triangle, the side BC and the hypo- thenuse have no common measure what- ever. If possible let D be a common measure of BC and AB; let BC con- tain D, m times, and let AB contain D mtimes. Let E be the square described on D. Then since A B contains D, m times, the square described on A B con- tains E,m X mor m? times. Similarly the square described on BC contains E mxXn or n* times: (Treatise on Geo- metry, Prop. 29.) But, because A B is an isosceles right-angled triangle, the square on A B is double of that on BC (Prop. 36,) whence m x m = 2n xX n or m* = 2n*. To prove the impossibility of this equation, (when m and 7 are whole numbers,) observe that m2 must be an even number, since it is twice the num- ber 2%. But m x m cannot be an even number unless m is an even number, since an odd number multiplied by itself producesan odd number *. Let (which | EE Stas 6 Sos DS TR * Every odd number, when divided by 2, gives a remainder 1, and is therefore of the form 2p +1, where p is a whole number. Multiply 2p + 1 by itself, which gives 4p2+ 4p +1, or2 (2p2-+- 2p) +1, which is an odd number, since, when divided by 2, it gives the quotient 2p2 + 2p, a whole number, and the remainder 1, 8] has been shewn to be even) be double of m' or m=2m!', Then 2m! x 2m! =2n?2 or 4m! = 2n* or n® = 2m’2, By repeating the same reasoning we show that 2 is even. Let it be 22’. Then, 2n/ x 2n! = 2m’? or m' = 2n'2, By the same rea- soning m' and n! are both even, and so on ad infinitum. ‘This reasoning shows that the whole numbers which satisfy the equation m? = 2m? (if such there be) are divisible by 2 without remainder, ad in- Jinitum. The absurdity of such a sup- position is manifest: there are then no such whole numbers, and consequently no common measure to B A and BC, Before proceeding any further, it will be necessary to establish the following proposition. If the greater of two lines A and B, be divided into m equal parts, and one of these parts be taken away; if the remainder be then divided into m equal parts, and one of them be taken away, and so on,—the remainder of the line A, shall in time become less than the line b, how small soever the line B may be. Take a line which is less than B; and eallit C. Itis evident that, by a con- tinual addition of the same quantity to C, this last will come in time to exceed A ; and still more will it do so, if the quantity added to C be increased at each step. To simplify the proof, we suppose that 20 is the number of equal parts into which A and its remainders are successively divided, so that 19 out of the 20 parts re- main after subtraction. Divide C into 19 equal parts and add to C a line equal to one of these parts. Let the length of C, so increased, be C’, Divide C’ into 19 equal parts and let C’, increased by its 19th part, be C’, Now since we add more and more each time to C, in forming C’, C’, &c., we shall in time exceed A. Let this have been done, and let D be the line so obtained, which is greater than A, Observe now that C’ contains 19, and C”, 20 of the same parts, whence C’ is made by dividing C” into 20 parts and removing one of them. The same of all the rest. Therefore we may return from D to C by dividing D into 20 parts, removing one of them, and repeating the process continually. But C is less than B by hypothesis. If then we can, by this process, reduce D below B, still more can we do so with A, which is less thanD, by the same method, This depends on the obvious truth, that if, at the end of any number of subtrac- tions (D being taken), we have left B55 B G 82 at the end of the same riumber of sub- tractions (A being taken), we shall have PA, since the method pursued in both q cases is the same. But since A is less than D, 2 A is less than 2. D, which q q becomes equal to C, therefore 2 A be- comes less than C *. q We now resume the isosceles right angled triangle. The linesBCandAB, which were there shown to have no common measure, are called zmcommen- surable quantities, and to their existence the theory of proportion owes its diffi- culties. We can nevertheless show that A and B being incommensurable, a line can be found as near to B as we please, either greater or less, which is commensurable with A. Let D be any line taken at pleasure, and there- fore as small as we please. Divide A into two equal parts, each of those parts into two equal parts, and so on. We shall thus at last finda part of A which is less than D. Let this part be E, and let it be contained m times in A. In the series E, 2E, 3E, &e., we shall arrive at last at two consecutive terms, pk and p + 1 E of which the first is less, and the second greater than B. Neither of these differs from B by so muchas E ; still less by so much as D; and both pE and p+ 1£E are commensurable with A, that is with mE, since Eis a common measure of both. If therefore A and B are incommensurable, a third magnitude can be found, either greater or less than B, differing from B by less than a given quantity, which magnitude shall be commensurable with A. We have seen that when A and B are incommensurable, there are no whole values of m and », which will satisfy the equation mA — xB = 0; nevertheless we can prove that values of m and 2 can * Algebraically, let a be the given line, and let — th part of the remainder be removed at every mM subtraction. The first quantity taken away is” m a and the remainder a— — or a(1— =) >» Whence ™m the second quantity removed is z. a— Ly; and m ms the remainder a— 2 (1 — es) ora(1— +) ., Se m m m milarly the nth remainder is a (1 — - Now m since 1— lis less than unity, its powers decrease, m and a power of so great an index ma be taken as +9 be less than any given quantity, i : STUDY OF, be found which will make mA = 7B less than any given magnitude C, of the same kind, how small soever it may be. Sup- pose, that for certain values of m and n*, we find mA — nB = E, and let the first multiple of E, which is greater than B, be pe, so that pE=B + E! where E/ is less than E, for were it greater, p —1 E, or pk — E, which is B-+ (E! — E), would be greater than B, which is against the supposition. The equation mA — nB = E gives pmA—-pnB=pE=B+E; whence pmA — (pn-+1)B =H let pm=m andpn-+1=”, whence mA—-n'B= EH’. We have therefore found a difference of multiples which is less than E. Let p' E’ be the first multiple of E/ which is greater than B, where p’ must be aé least as great as p, since E being greater than E’, it cannot take more of E than of E’ to exceed B. Let — piE’ =B-+ E", then, as before, mplA —- n'p' +1 B=E", or m'A = n’B = XK’; we have therefore still further diminished the difference of the multiples; and the process may be repeated any number of times ; it only remains to show that the diminution may proceed to any extent, This will appear superfluous to the beginner, who will probably imagine that a quantity diminished at every step, must, by continuing the number of steps, at last become as small as we please. Ne- vertheless if any number, as 10, be taken and its square root extracted, and the square root of that square root, and so on, the result will not be so small as unity, although ten million of square roots should have been extracted. Here is a case of continual diminution, in which the diminution is not without limit. Again, from the point D in the line | AB draw DE, making an angle with | A Bless than half a right angle. Draw BE perpendicular to AB, and take) * It is necessary here to observe, that in speak- ing of the expression mA —”B we more frequently © refer to its form, than to any actual value of it, | derived from supposing m and n to have certain — known values. When we say that mA—nB can | be made smaller than C, we mean that some values | can be given to m and » such that mA—nB = x n! m m® Therefore (1) 2 nl ™m® Pe > =Ha< nl mm me | n! sa st >=< i) | m Therefore (2) mA) _ n' Bea mi Pt > = S{ntQ i Or, if four magnitudes are proportional, according to the common notion, it fol lows that the same multiples of the firs! and third being taken, and also of th: second and fourth, the multiple of th’ first is greater than, equal to,or less than | : MATHEMATICS. that of the second, according as that of the third is greater than, equal to, or less than, that of the fourth. This property* necessarily follows from the equations mA —nB=0 mP —nQ = 0’ follow that the equations are necessary consequences of the property, since the latter may possibly be true of incom- mensurable quantities, of which, by deti- nilion, the former is not. The existence of this property is Euclid’s definition of proportion: he says, let four magni- tudes, two and two, of the same kind, be called proportional, when, if equimulti- ples be taken of the first and third, &e., repeating the property just enunciated. What is lost and gained by adopting Euclid’s definition may be very simply stated; the gain is an entire freedom from all the difficulties of incommensu- rable quantities, and even from the neces- sity of inquiring into the fact of their existence, and the removal of the inac- curacy attending the supposition that, of two quantities of the same kind, each is a determinate arithmetical fraction of the other ; on the other hand, there is no obvious connexion between Euclid’s de- but: it does not therefore _ finition and the ordinary and well-esta- blished ideas of proportion; the defini- tion ifself is made to involve the idea of _ infinity, since all possible multiples of the four quantities enter into it; and lastly, the very existence of the four quantities, called proportional, is matter for subsequent demonstration, since to a beginner it cannot but appear very un- likely that there are any magnitudes which satisfy the definition. The last objection is not very strong, since the learner could read the first proposition of the sixth book immediately after the definition, and would thereby be con- vinced of the existence of proportionals ; the rest may be removed by shewing another definition, more in consonance with common ideas, and demonstrating that, if four magnitudes fall under either of these definitions, they fall under the other also. The definition which we propose is as follows:—* Four magni- tudes, A, B, P, and Q, of which B is of the same kind as A, and Q as P, are said to be proportional, if magnitudes B+C and Q+R can be found as near as we please to B and Q, so that A, B+C, P and Q+R, are proportional accord- ing to the common notion, that is, if * It would be expressed algebraically by saying, that if mA-—2B and mP—nQ are nothing for the same values of m and n, they are either hoth positive or both negative, for every other value of m and n, 85 whole numbers ’m and 7 can satisfy the equations mA —2(B+C)=0 mP —n(Q+R)=0. We have now to shew that Euclid’s de- finition follows from the one just given, and also that the last follows from Eu- clid, that is, if there are four magnitudes which fall under either definition, they fall under the other also. Let us first sup- pose that Euclid’s definition is true of A, B, P, and Q, so that mA nB mes eS 1nQ This being true, if will follow that we can take m and 7, so as not only to make mA —nB less than a given magnitude E, which may beas small as we please, but also so that mP—xnQ@ shall at the same time be less than a given magnitude F, however small this last may be. For if not, while m and » are so taken as to make mA-—m7B less than E (which it has been proved can be done, however small E may be) suppose, if possible, that the same values of m and 7 will never make mP—n7Q less than some certain quantity F, and let pF be the first mul- tiple of F which exceeds Q, and also let E betaken so small that pE shall be less than B, still more then shall p¢mA — 7B), or pmA—pnB be less than B. But since pF is greater than Q, and mP —nQ is by hypothesis greater than F, still more shall mpP—npQ be greater than Q. We have -then, -if our last supposition be correct, some value of mp and np, for which mpA —npB is less than B, while mpP —npQ is greater than Q, or ; mpA is less than (zp+1)B, mpP is greater than (np+1)Q, which is contrary to our first hypothesis respecting A, B, P, and Q, that hypo- thesis being Euclid’s definition of pro- portion, from which if mp is less than xp-+1 B mpP is less than np--1 Q. We must therefore conclude that if the four quantities A, B, P, and Q, satisfy Euclid’§ definition of proportion, then m and 2 may be so taken that mA—nB and mP—nQ shall be as small as we . please. Let mA —nB=E and E=nC mP—nQ=F Hs ret: Then mA—n(B-+C)=0 mP —n(Q-+R)=0, and since E and F can, by properly as- suming m and m, be made as small as we please, much more can the same be 86 done with C and R, consequently we can produce B+C and Q+R as near as we please to B and Q, and propor- tional to A and P, according to the com- mon arithmetical notion. In the same way it may be proved, that on the same hypothesis B — Cand Q — Rean be found as near to B and Q as we please, and so that A, B-C, P and Q -R, are propor- tional according to the ordinary notion, It only remains to shew that if the last- mentioned property be assumed, Euclid’s definition of proportion will follow from it, That is, if quantities can be exhi- bited as near to P and Q as we please, which are proportional to A and B, ac- cording to the ordinary notion, it follows that mA. nB ne | ne in (70. For let B+-C and Q-+R be two quan- tities, such that fA-g(B+C)=0 in which, by the hypothesis, f and g can be so taken that C and R are as small as we please. We have already shewn that in this) case (mand 2 being any numbers whatever) mA is never greater or less than 2(B+C), without mP hbe- ing at the same time the same with re- gard toz(Q+R). That is, if mA. is greater than nB+nC, then MP. eee » 2Q+nR, Take some given* values for m and n, fulfilling the first condition; then, since C and R may be as small as we please, the same is true of 2C and mR; if then mA is greater than 2B ys BUR Oe opt De Bopha rene URAL arc] 6 For if not, let mA=nB+ 2, while mP= nQ —-Y, © and y being some definite magnitudes. Then if nB-+-x2 >nB--nC nQ —-y>nQ-+nR, which last equation is evidently impos- sible ; therefore if mA>nB, mP>nQ. In the same way it may be proved, that if mA 4 then mq > np, or a >= mp —-, or = i anda oe P m7 2 pir ie 3 n tin . ret less than unity, and 1 therefore : P any fraction multiplied by this, is dimi- m+n 35 it 1 += P+4a p. ‘tv Pp therefore less than > the greater of the nished. But and is two. In the same way it may be proved to be greater than _ the least of the two, . . . . b! This being premised, since — q!! b' all +b : b! al! b t + a I t lies between —— and — or a’ a! a ail a b' b between —- and a a Call the coefficients of A and B in the series of equations, a de, &e. by be &e. and form the series of fractions dS Ba Os &e. Ao U3 2 The two first of these ay will be — and ~a and X > X’, whence 2aX! << wWX+ aX’, or 2aX’ < B and xe = the remainder therefore which a ° a . . comes in the +1 equation is less - than the part of B arising from dividing it into twice as many equal parts as {here are units in the 2" coefficient of A; and as this number of units may increase to any amount whatever, by earrying the process far enough, 2a “may be made as small as we please, and, a fortiori, the remainders may be made as small as we please. The same theorem may be proved in a similar way, if we begin at an even step of the pro- cess. Resuming the equations aKa = be Be -p! eds From the second, A = —B - pt / and since X’ < ee b' B — B, or is too small. Again, by yesh se = Se Boe Agee Now a a a a D.C. X’ a; whence Vo ai that US is, 5 B exceeds A by a less quantity than b e * B falls short of if, so that - is a nearer representation of A than 3 though on a different side of if. We have thus shewn how to find the representation of a line by means of a linear unit, which is incommensurable with it, to any degree of nearness which we please. This, though little used in practice, is necessary to the theory; and the student will see that the method here followed is nearly the same as that of continued fractions in Algebra. We now come tothe measurement of an angle; and here it must be observed that there are two distinct measures em- ployed, one exclusively in theory, and one in practice. -The latter is the well- known division of the right angle into 90 equal parts, each of which is one degree ; that of the degree into 60 equal parts, each of which is one minute; and of the minute into 60 parts, each of which is one second. On these it is unnecessary to enlarge, as this divi- sion is perfectly arbitrary, and no reason can be assigned, as far as theory is con- cerned, for conceiving the right angle to be so divided. But it is far otherwise with the measure which we come to consider, to which we shall be naturally led by the theorems relating to the circle. Assume any angle, AOB, as the angu- lar unit, and any other angle, AOC, Let r be the number * of linear units con- * It must be recollected that the word number means both whole and fractional number, 90 tained in the radius OA, and ¢ and s the lengths, or number of units contained in the arcs AB and AC. Then since the angles AOB and AOC are propor- tional to the ares AB and AC, or to the numbers ¢ and s, we have Angle AOC is 7 of the angle AOB; and the angle AOB being the angular unif, the number > is that which ex- presses the angle AOC. This number is the same for the same angle, what- ever circle is chosen; in the circle FD the proportion of the ares DE and DF is the same as that of AB and AC: for since similar ares of different circles are proportional to their radii, AB ; DE ;; OA; OD Also AC:DF::0A:OD eh: dans eek Palle therefore the proportion of DF to DE is that of s to z, and - is the measure of the angle DOF, DOE being the unit, as before. It only remains to choose the angular unit AOB, and here that angle naturally presents itself, whose arc is equal to the radius in length. This, from what is proved in Geometry, will be the same forall cireles, since in two circles, arcs which have the same ratio (in this case that of equality) to theirradii, subtend the same angle, Let ¢=~7, then < is the number corresponding to the angle whose arc is s. This is the number which is always employed in theory as the measure of an angle, and it has the advantage of being independent of all linear units ; for suppose s and7 to be expressed, for example, in feet, then 125 and 127 are the numbers of inches in the same lines, and by the common theory 1 of fractions = = —— Generally, the al- teration of the unit does not affect the number which expresses the ratio of two magnitudes. When it is said that the Bae is angle = That it is only meant that, on one particular supposition, namely, that the angle 1 is that angle whose arc STUDY OF is equal to the radius, the number of these watts in any other angle is found by dividing the number of Zznear units in its are by the number of linear units in the radius. It only remains to give a formula for finding the number of degrees, minutes, and seconds in an angle, whose theoretical measure is given. It is proved in geometry that the ratio of the circumference of a circle to its diameter, or that of half the circumfe- rence to ifs radius, though it cannot be expressed exactly, is between 3.14159265 and 3.14159266. ‘Taking the last of these, which will be more than a suf- ficient approximation for our purpose, if follows that the radius being 7, one half of the circumference is 7 x 3.14159266$ and one-fourth of the circumference, or the are of a right angle, is r x 1.57079633. Hence the number of units above described, in a right angle, is Be » or 1.57079633. And the num- radius ber of seconds in a right angle is 90x 60x60, or 324000. Hence if $ be an angle expressed in units of the first kind, and A the number of seconds in the same angle, the proportion of A to 324000 will also be that of 9 to 1.57079633. To understand this, recol- lect that the proportion of any angle to the right angle is not altered by chang- ing the units in which both are expressed, so that the numbers which express the two for one unit, are proportional to the like numbers for another, Hence A : 324000 :: 9: 1.57079633 ; 324000 A. 1.57079633 Soa or A = 206265 x 9, very nearly. Suppose, for example, the number of seconds in the theoretical unit itself is required. Here 3 = land A = 206265; 1 imilarly i 19= —— similarly if A be 1, 9 OdaEE> the expression for the angle of one se- cond referred to the other unit. In this way, any angle, whose number of seconds Is given, may be expressed in terms of the angle, whose are is equal to the ra- dius, which, for distinction, might be called the theoretical unit. This unit is used without exception in analysis ; thus, in the formula, for what is called in trigonometry the sine of x, viz., 5] —, &e. or A = which is 206265" MATHEMATICS. ’ The number 3.14159265, &c. is called #, and is the measure, in theoretical units, of two right angles. Also-is the “measure of one right angle ; but it must not be confounded, as is frequently done, with 90°. It is true that they stand for the same angle, but on different sup- positions with respect to the unit; the unit of the first being very nearly 206265 aa times that of the second. There are methods of ascertaining the value of one magnitude by means of another, which, though it varies with the first, is not a measure of it, since the increments of the two are not propor- tional ; for example, when, if the first be doubled, the second, though it changes in a definite manner, is not doubled. Such is the connexion between a num- ber andits common logarithm, which latter increases much more slowly than its number ; since, while the logarithm changes from 0 to 1, and from 1 fo 2, the number changes from 1 to 10, and from 10 to 100, and so on. . Now, of all triangles which have the same angles, the proportions of the sides are the same. If, therefore, any angle pf cH 91 B, B’, B’, &e. in one of its sides, and 3, b!, &e. in the other, perpendiculars be let fall on the remaining side, the tri- angles BAC, B/AC’, bAc, &c. having a right angle in all, and the angle A com- mon, are equiangular ; that is, one angle being given, which is not a right angle, the proportions of every right-angled triangle in’ which that angle occurs are given also; and, vice versa, if the pro- portion, or ratio of any two sides of a right-angled triangle are given, the angles of the triangle are given (Geom. Li, 32>): To these ratios names are given; and as the ratios themselves are connected with the angles, so that one of either set being given, véz. ratios or angles, all of both are known, their names bear in them the name of the angle to which they are supposed tobe referred. Thus, a or Bidveupaalts to ey is called the sine AB ., AC — side opposite to B iF Nati AB’ °* hypothenuse or the sine of B, the complement * of A, is called the cosine of A. The following table expresses the names which are i to th ANd ectibak Moki ay: given to the atios, TB AB AG AC AB AB ; BG AG and BO’ relatively to both angles, with the abbreviations made use of. The terms opp., adj., and hyp., stand for, opposite side, adjacent side, and hy- pothenuse, and refer to the angle last hypothenuse CAB be given, and from any points mentioned in the table. we , is the mie a bans Thee are written, | AB sine of A pedi cosine of B Bae sin A cos B ae cosine of A i sine of B | ; an cos A sin B a tangent of A ane cotangent of B 7 tan A cot B = cotangent of A ot. Paced of B att cot A fan B Bs secant of A ae cosecant of B ae: sec Al cosec B a cosecant of A a secant of B af cosec A Hed B * When two angles are together equal to a right angle, each is called the complement of the other, Generally, complement is the name given to one part of a whole relatively to the rest. Thus, 10 being made of 7 and 3, 7is the complement of 3 go 10, 92 Tf all angles be taken, beginning from one minute, and proceeding through 2’, 3’, &e., up to 45°, or 2700’, and tables be formed by a calculation, the nature of which we cannot explain here, of their sines, cosines, and tangents, or of the logarithms of these, the proportions of every right-angled triangle, one of whose angles is an exact number of minutes, are registered. We say sines, cosines, and tangents only, because it is evident, from the table above made, that the co- secant, secant, and cotangent of any angle, are the reciprocals of its sine, co- sine, and tangent, respectively. Again, the table need only include 45°, instead of the whole right angle, because, the sine of an angle above 45° being the co- sine of its complement, which is less than 45°, is already registered. Now, as all rectilinear figures can be divided into triangles, and every triangle is either right-angled, or the sum or difference of two right-angled triangles, a table of this sortis ultimately a register of the pro- portions of all figures whatsoever. The rules for applying these tables form the subject of trigonometry, which is one of the great branches of the application of algebra to geometry. In a right-angled triangle, whose angles do not contain an exact number of minutes, the propor- tions may be found from the tables by the method explained in Chapter XI. of this treatise. It must be observed, that the sine, cosine, &c. are not measures of their angle; for, though the angle is given when either of them is given, yet, if the angle be increased in any propor- tion, the sineis not increased in the same proportion. Thus, sin 2A is not double ofsin A. , The measurement of surfaces may be reduced to the measurement of rect- angles; since every figure may be di- vided into triangles, and every triangle is half ofa rectangle on the same base and altitude. The superficial unit or quan- tity of space, in terms of which it is chosen to express all other spaces, is per- fectly arbitrary; nevertheless, a com- mon theorem points out the convenience of choosing, as the superficial unit, the square on that line which is chosen as the linear unit. If the sides of a rect- angle contain a and 6 units (Geometry I, 29.), the rectangle itself contains ad of the squares described on the unit. This proposition is true, even when a and b are fractional. Let the number of units in the sides be-- and mi and take ano- STUDY OF ther unit which is os of the first, or is ob- tained by dividing the first unit into 2q parts, and taking one of them. Then, by the proposition just quoted, the square described on the larger unit contains nq X nq of that described on the smaller. ee m Again, since a and = are the same fractions as —- and ae they are form- nq nq ed by dividing the first unit into mq parts, and taking one of these parts mq and np times; that is, they contain mq and np of the smaller unit ; and, therefore, the rectangle contained by them, con- tains mq X mp of the square described on the smaller unit. But of these there are mq X mq in the square on the longer unit; and, therefore, maxn y poke Mir oes fhigy ios ae ng X nq ny nq nq ber of the larger squares contained in the rectangle. But aq the algebra- ical product of aand n q This propo- sition is true in the following sense, where the sides of the rectangle are in- commensurable with the unit. What- ever the unit may be, we have shewn that, for any incommensurable mag- nitude, we can go on finding 6 and a two whole numbers, so that , read — 29 20, 59 14, jor 4 a b*, read 4 a® b?, oa 28, 35, 96 trom. bottom, for ba, read bu’. 10" ss os ope be for + 6, read — 6°. gets ays eee 95 for — 64, read + G6. » 33, 5, 3 * for equation read equations. » 34, 5, 3 a4 for a, read d. » 39, 5 13 for + a, read + a’. 39 om 42, es’) th, for ce.erend a. 5, 46, ,, 23 and 26, for greater, read less. 55 47, 5, end of 2, for a full stop, read a comma, —, 5 from bottom, for J 49 — 24, read J 49 84, —, 1 from bottom, for ,/221 + 9, read AV 221 + 9. 3 49, 21,.for +.9, read + 11. » 52, 5, for 1 —, read — 1. p my p » 95, 7, for (a) , read (4 ) , > —, | from bottom, for NA a’, read °/ i% ay Os ed, fOr" Ni ane read "Va “ae, 5 —> 1 from bottom, for Sey ‘): ahs LEA ay 55 —, 22 from bottom, for ey read —” g mp que 9 o—, 22 ‘5 » for —-, rea na ng 55 —, 6 from bottom, for subtracted, read subtraction. », 98, 3 and 4 from bottom, for a +A™, reada + is » 60, 3 from bottom, for a, read a*, ARITHMETIC AND ALGEBRA. Notation and Definitions. 1. Iris by means of numbers that we are able to express the magnitude, that is, the size, of any thing, or of any col- lection of things that are of the same kind. To do this in any case, we first fix on some portion of that kind of thing, the magnitude of which portion is well known; we then state the num- ber of times that this portion is con- tained in the thing or collection of things in question. When, for instance, we say that a distance is twelve miles, or a weight twelve pounds, we state how often the known portion of dis- tance, one mile, or the known portion of weight, one pound, is contamed in the distance or the weight of which we are speaking. ‘This portion so fixed on is called a unit. In the examples just referred to, a mile is used as the unit of distance; a pound, as the unit of weight. In expressing the magnitude of anything, or in subjecting it to calcu- lation, we may make use of that unit which we may find most convenient. We may express a distance in miles, yards, feet, or inches, or a weight in pounds, ounces, or grains. But when we have once fixed on a unit, we must keep its nature clearly in view, other- wise we shall often fall into errors. Sometimes there is something in the nature of the thing whose magnitude is to be expressed, that furnishes us with a unit by which to express it. In stating the size of a crowd of people, or a fleet of ships, every body would employ a single person, or a single ship, as a unit, and would say how many people or how many ships there were. Some- times, again, there is nothing to make us choose one portion more than an- other. In the cases of weights and measures, for instance, the units are fixed only by an understanding among the community who use them; such units are said to be arbitrary. It is often yery conyenient that the same Q) unit should be used as extensively as possible. On this account most govern- | ments have made laws enjoining a uniformity of weights and measures, | and men of science have invented means of finding the standard unit at all places and times. When we consider a num- ber in the abstract, as when we say that 10 and 6 make 16, the number 1 is our unit. It is called unzty. 2. A portion definite or indefinite, known or unknown, of any thing that can be expressed by means of units, is called a quantity ; and may be made the subject of the operations of arith- metic or algebra. In arithmetic, we only deai with quantities expressed in numbers, which, consequently, are al- ways known and definite. But in al- gebra it is very often necessary to con- sider quantities that are indefinite or unknown. These quantities cannot pos- sibly be expressed by numbers, for if they were they would be no longer in- definite ; and if we do not know them, how can we find numbers to express them? ‘To represent such quantities, the letters of the alphabet are used. Arithmetic and algebra are thus very intimately connected. In treating of them together, we shall find, on the one hand, that the notation of algebra is very useful in explaining the operations of arithmetic; and, on the other, that these operations furnish perhaps the best practical illustrations of the results of algebra. 3. A man can dig a piece of ground in ten days which it takes his son six- teen days to dig ; in what time can they dig it if they work together? The time sought, though it be unknown, is yet a fixed definite number of days; a num- ber capable of being halved or doubled, multiplied, divided, or submitted to any other operation that can be performed on known quantities. We cannot de- note it by a number, for we do not know what number to take ; when we are going to operate on it, then, which B 2 ARITHMETIC AND ALGEBRA. we must do before we can find what it is, we denote it by a letter; 2, for in- stance. a is properly called an wn- known quantity. Again ; the time in which the father and son can dig the field working to- gether, depends on the time in which the father can dig it alone, and the time in which the son can dig: it alone, and on nothing else. Suppose that we wish to find generally, and without re- ference to any particular instance, in what way the time sought depends on the father’s time and on the son’s time. If we express the time taken by the father and that taken by the son in numbers, we may find the time taken by both together ; but then it will bea number also, and we shall have an- swered the question for one particular case only; we shall not have found the general relation which we have seen must subsist. But if we call the time taken by the father a, and that taken by the son 4, we do find a general re- lation if we can in any way find the time taken by both together, since a may be any number of days, and 6 may be any number of days. Here a and 6 are quantities that are indefinite, that is, quantities that may stand for any numbers whatever. But they are called known quantities, because they must be known before we can answer the ques- tion in any particular case. It is usual to denote known quantities by the first letters, as a, 6, c, &c.; and unknown by the last, as a, y, z. 4, When we have any number of quantities, and wish to find or to ex- press how many units are contained in them all taken together, we do it by addition. When quantities are to be added together, we write them with the mark + between them. Thus5+7+11 shows that 5, 7, and 11 are to be added together ; and when this is done, the result is called the swm of these num- bers. Soa-+ 6+ cis the sum of a, b, and c, expressing how many units are contained in these quantities taken to- gether. + is read plus (a latin word meaning more) ; it is called the sign of addition, or the positive sign; and a quantity to which it is prefixed, is called a positwe quaniity. 5. When we have two quantities, and wish to find or to express the num-. ber of units,in one of-them, after the number of units in the other has been taken away trom it, we do this by sud- traction, When one quantity is to be -that 6 is to be subtracted from a. | subtracted from another, we write it after the other with the mark — be- | tween them. Thus 9 — 5 shows that 5 is to be taken from 9, and is called the difference of these numbers. a— b is the difference of a and 6, and shews is read minus (a latin word meaning less); it is called the sign of subtrac- tion, or the negative sign, and a quan- tity to which it is prefixed is called a negative quantity. 6. When a quantity has neither of these signs annexed to it, + is always supposed to be its sign, and it is a positive quantity. 7. It will often happen, that in adding quantities together we have to add the same quantity to itself once or more. We may, for example, have such sums as 5 +5+.5, or atatart &e., where a may be supposed to be written 5 times, or as often as there are unities in 6, The process by which we find such sums, or by which we express this continued addition, is called multiplt- cation. To add three fives together is called multiplying 5 by 3; to take a as often as there are unities in 6 and add them all together, is called multiplying a by 6. These processes we express by writing 3 xX 5 or 3.5; 6 x a, 6.a, or more frequently 6a. 6a is called the product ot banda; 6 and a are called the factors of ba. 6 is the multiplier, and ais the multiplicand. Soa. bc is the product of a and be. 8. We very often have such a pro- duct as aa, or a multiplied by itself. This is written a2. So aaais written a. Thus 2? is 2.2.2, or 8, and 164 is 10.10.10.10, or 10,000. The num- ber written above to the right, shows the number of times that the quantity to which it is annexed is a factor, and is called the index, or the exponent of that quantity. a?, a5, &e. are called powers of a. a® is the second power, or square of a; a? is its third power, or cube ; and a) is its fifth power. So a” is the m” power of a; meaning that a is a factor as olten as there are unities in m; whatever number m may be. -9. When there is a product such as 3a, or 5 a2c, where one of the factors -1s a number, that number is called a coefficient ; 3 is the coefficient of a in. the expression 3 a, and 5 that of a in 5a?c. When the coefficient is unity, itis not usual to write it; thus 16¢ and 6 ¢ are the same. 10, Ifthere be two quantities, aand 4, “\ ARITHMETIC AND ALGEBRA, 3 and it be required to find a third quan- tity, such that when multiplied by 6 the product shall be a, we do this or express it by division. To divide 20 by 4, for instance, is to find a number which, when multiplied by 4, will give 20 for product. This number we know to be 5. Here 20 is called the dividend, 4 the divisor, and 5 the quotient. We express that a is to be divided by 6 by writing a6, or more frequently 33 where a is the dividend, 6 the divisor, and ; the result of the division, or the 3 VLU 20 quotient. Similarly, ri or 20—4 means 5. It will be seen at once, that this ac- count of division corresponds with the uses to which we have been accustomed to apply it in arithmetic. Thus, if we want to find how many shillings there are in 180 pence, we divide 180 by 12. Now, whatever be the number of shil- lings sought, since there are 12 pence in a shilling, 12 times that number must make 180; so that the number required is the number which, when multiplied by 12, gives 180 for product. Again, if there be 725 men, and we wish to di- vide them into 25 equal companies, and for that purpose to find how many men are to be in every company; 25 times this number, whatever it may be, must make 725. Weseek the number then, which, when multiplied by 25, gives 725 for product, we find it by dividing 725 by 25. ae As multiplication is a continued ad- dition, division may be regarded as a continued subtraction. In dividing 30 ‘by 6, for example, we may be said to find out how often 6 can be subtracted from 30; that is, how often6 is con- tained in 30. 11. The square root of any quantity, a, is a quantity which, when multiplied by itself, produces a. We express the square root of a by writing ?,/a, or simply s/a. Thus 4/4 is 2, for 2.2 is 4; /b2 is b, for 6.6 makes 62. The cube root of ais a quantity which, when multiplied into itself twice, pro- duces a; we express it by writing °,/a. Tras PQ/2 7sS182 foe 343%, 8. Ie a a sO 4,/a is a quantity which, when multi- plied into, itself 3 times, produces a, and is called the fourth root of a. Similarly, 5,/a is the fifth root of a, and so on; and ™,/a is the m” root of a, or that quantity which, multiplied by itself as often as there are unities in the number which is one less than m, produces a. The mark 4/ is called the radical sign, from a Latin word mean- ing a root. 12. When a quantity is made up of other quantities connected with each other by means of the signs + and —, the quantities of which it is so com- posed are called its terms. Thus ac, —ab, and ad?, are the terms of ac —ab-+ ad, 13. When any of the operations we have been describing is to be performed on a quantity consisting of more than a single term, to prevent ambiguity the quantity is enclosed in brackets, or has a line drawn over it. Thusa —(e+d +e), ora—c+d+e, means that the whole quantity ec + d + eis to be taken from a; whereas if it were written a — c+ d+ e, it would mean that c only is to be taken from a, and d and e added toit. So (a+ 6) (e+ ad), or at+éd. e+d means that the whole quantity ce -+ dis to be multiplied by the whole quantity a+ 6. The line drawn overa quantity for this purpose is called a vinculum, (the Latin word for a fetter) or.a bar. In the same way (a? + b?)8 means the cube of a2 + 62, 14, The mark = is called the sign of equality ; it shows that the quantities between which it stands are equal to one another. Thus 7+ 8— 9 = 6, 9X 13=117. As further examples of what has been laid down, make a= 1, b = 2,¢=38, and d=4, and we shall find that 7at+7e—-3b—d=74+21-6-—4=18, (Qa+ bc) (4d —bd) = (2+ 6).(016 — 8) = 8.8 = 64, cht ed «2412 14 2¢e—-a 6—4 2 (502 — 3) @ = @ —b)? = (5 —3) 16 — (4 —2)9= 32 —8 = 24, B 2 -Of Addition and Subtraction.. 15. When quantities, if they differ at all, differ only in their coefficients [art. 9] they are called dike quantities. When they differ in other respects they are called unlike quantities. 2acand 3ac, 5 aand7 a?, 27 a b and a3 b are severally couples of like quantities ; a, 6,5 a2, 5a2b,3ab are all unlike quantities. When we have to add, or to subtract unlike quantities, we write them in the same line with the sign of addition or of subtraction between them. ‘Thus a added to 5 we can express only by writing a+b; ab? subtracted from a® 6 we express by writing a? 6 — a bz, But take the case of like quantities, and suppose that 2a is to be added to 3a. That is, a+ a is to be added to at+a+ta [art.7]. The sum must be atatata+ta, or (3 + 2) a, or 5a. So 15 a3 + a = (15 + 1) a8 [art. 9] or 16 a’. In the same way we find that 3@—3a= (5 — 3) a, or 2a; and 27 @&b — 15a8b = (27 — 15) a& B, or 12 a8 b, There is no difficulty then in bringing together by means of addition and sub- traction any number of quantities con- sisting each of one term. Those that are like are to be collected into one term by adding or subtracting their co- efficients, according as their signs are positive or negative [arts.4 and 5]; and this term is to be set down with the sign of addition or subtraction, according as the positive or negative quantities col- lected into it are greater. If the nega- tive be greater, the quantities to be sub- tracted are together greater than those to be added, and therefore the single term made up of them allis to be subtracted. Those quantities that are unlike are to be set down, each with its proper sign. For example, e+ 25a6?—10a2 —8r7b24+ 1742 = (1 — 10 +17) a? + 25a 02 —8ax 2 = 8a? + 25a 62 — 8 x22, 16. In the same way we find that 38a+b—5a+6a—b—Il6a =—12a., ARITHMETIC AND ALGEBRA. Here the whole can be collected into a single term which is negative. In algebra we very often come to a nega- tive result. We may, perhaps, find it) difficult to attach any meaning to a. negative quantity in the abstract, a quantity to be subtracted without any thing to subtract it from. But, when- ever we have a result of this kind, we shall always find something in the na- ture of the question we are considering that enables us to give the negative sign ameaning. Of this we shall have many instances as we proceed ; to take a familiar one now, suppose a and 6 in the last example to be sums of money, and that a man is taking an estimate of his outstanding accounts, if he set down sums due to him with -+ before them, he should set down sums due b him with —. When he takes the amount, if his debts be the greater, of course the balance will have — before it; with respect to these accounts he has so much less than nothing. We shall find, that when algebra is applied to questions that occur, there is no such thing as a quantity essen- tially negative. It is negative only because we choose so to consider it; and we may make it positive, if at the same time we make negative the quan- tities formerly positive. In the in- stance just given, the man may con- sider his debts as positive, and his cre- dits to be subtracted from them as ne- gative. On this supposition his result would have had the positive sign; but this would not at all have changed its nature, a positive debt and a negative credit beimg algebraically the same thing. 17, We now come to the addition and subtraction of quantities that con- sist of more than a single term. To add to a the quantity 6 + ¢, is to add 6 and c both; the sum therefore is a+6b+c. To add to a the quantity 6 — c, is to add 6 diminished by ¢; the sum therefore is a+ b—c. We find then, that to add a quantity consisting of more than a single term, is to take ats terms one by one, adding those that are positive and subtracting those that are negative. 18. To subtract 6 +c from a, is to subtract 6 and c both; the result there-. fore isa —6—c. Insubtracting 6 —¢ from a, if we subtract 6 aione from a, and write a — 6, we shall have subtracted too much; for it is not 6, but d dimi- nished by ¢, that is to be subtracted. ARITHMETIC AND ALGEBRA, 5 We must therefore add ¢ toa—b to have the proper result, which is a—b+e. Our two last results, which are very important, are a—(b+c=a-—b—e, and a—(b—c)=a—brte. It is plain that these apply equally to quantities of more than two terms. To subtract a quantity then of more than one term, we must change the sign of every one of its terms from + to —, or Srom — to-+, and then proceed as if the quantity so altered were to be added. To subtract a negative quantity is equi- valent to adding a positive. __A single example in numbers will illustrate this: 10—(8'—-5) = 10—84+5=7 by the rule: again, since 8 — 5 = 3, 10 — (8 —5) = 10 —3, or 7, of course the same result as the rule gave, Of these operations take the following examples ; 1 3@+4b6c~@+ (5a°—6bc—15) + 21-4 a2 — 102), the sum is 4@2—2be—1l1lea@+t 6, 2, 5a@+4ab—6xay—- (lle +6ab—4zy), the difference is —6a?—2ab—2xy, 3, 4a—3b—(5c-a—5b—e) + (10~7a + 30), the suin is —2a+2b—5cec4H4e+ 10. In this last example, before proceeding to collect the terms together, we ob- serve that the quantities within the brackets are respectively 5c—a— 55+ e,and10—7a—3e. 19. In explaining the reasons of the rules for addition and subtraction in arithmetic we must first observe, that all numbers are represented by means of the nine digits, as the symbols for the first nine numbers are called, and zero, or the symbol for nothing. This is done by agreeing that the value of each digit shall be ten times as great as it would have been if it had held the next place to the right. The first digit to the right stands for so many units, the one oext it for so many tens, the next for so many hundreds, and soon. A number chen such as 57624 is equivalent to | 50000 + 7000 + 600 + 20+ 4. | 20. The common rule for the addi- ‘ion of numbers is this: Write the wmbers to be added under one another, ‘0 that the units’ digits shall be all in me column, the tens’ all in another, the wndreds’ allin another, and so on ; cast ip the units’ column, and write under t the units’ digit of its sum, carrying he tens’ digit to be added to the next olumn ; cast up the tens’ column with his addition, write under it the units’ hgit of its sum, carrying the tens’ digit 0 be added to the next column ; proceed hus with every column till the last, mder which write its whole sum, The reason of this rule will appear at once by stating an example of it*in detail, 4765 4000 + 700+ 60+ 5 8904 8000 + 900 +4 725 700 + 20+ 5 7500 7000 + 500 21894 19000+ 2800 +80 +14 The first result is found by the rule; the other is found by casting up each column separately, and is the same as 19000 + 2000 + 800 + 80+ 10+ 4, Collecting the numbers of the same denomination into one, which is what we do when we carry as directed by the rule, this becomes 21000 + 800 + 90+ 4, or 21894. 21. The rule for subtracting one number from another is this: Write the number to be subtracted under the other, as in addition ; subiract every digit in the lower number from the one over tt in the upper, and write the remainder under it. When the digit in the upper number is the smaller, add ten to itt, observing at the same time to increase the digit next to tts left in the lower number by unity. To prove this, suppose the two num- bers to be 8437 and 5974. If we write them thus 8000 + 400 + 30+ 7, 5000 + 900+ 70 +4, 6 ARITHMETIC AND ALGEBRA. and proceed to subtract them term from term, we find that we cannot take 70 from 30, or 900 from 400, without having negative quantities appearing in our result, which we wish to avoid. We therefore write them thus 7000 + 1300 + 130 +7, 5000 + 900+ 70+ 4, and then, subtracting them term from term, we find for their difference 2000 + 400 + 60 + 3, or 2463. It is manifestly the same thing to in- crease a term in the lower number by unity, as to diminish the corresponding one in the upper by unity, and this consideration brings us to the rule. 22. There is a way in use, of subtract- ing numbers by means of what is called their arithmetical complement. Sup- pose, for instance, that we have to sub- tract 817 from 2573; that is, to find the difference 2573 — 817. This difference will not be affected by adding any quan- tity to it, if at the same time we sub- tract the same quantity from it. Let this quantity be 1000 ; 2573 — 817 then is the same as 2573 + (1000 — 817) — 1000, or as 2573 + 183 — 1000, or as 2573 + 1183, if we observe in adding the two numbers to subtract the 1 in respect of the negative sign placed above it; this subtraction from the place that I occupies being equivalent to the subtraction of 1000. 1183 is called the arithmetical complement of 817. As we can perform such a sub- traction as 817 from 1000 mentally, and write down the result almost as fast as we could write 817 itself, the arithme- tical complement may furnish us with amore expeditious way of taking the balance of such a set of numbers as 2573 — 183 + 17856 — 1273 + 534, than taking the separate sums of the positive and negative terms, and then finding the difference of these sums. By arithmetical complements we have 2573 1817 17856 18727 — ~634 19507. This way of writing numbers with a negative unit in’ the left hand digit’s place is also in use in logarithms. Of Multiplication. 23. To multiply 6 by awe write ad, and to multiply a by b we write ba, ab and ba are equal to each other, just as4 x5 and 5 x 4 are equal to each other, both being 20. Of the truth of this we may satisfy ourselves thus, a6 means that the units in 0 are to be taken as often as there are units in a, [art. 7]. : 1+1+1+1+1 I1+1+14+1 © 1+1+1+1+1 1+1+1+41 t+1+1+1+1 1+14+1+1 1+1+14+1+4+1 1+1+1+1 24 1+14+1+1 Now write 1 as many times in the same line as there are units in 0, and make as many such lines as there are units ina. The value of each line is }, and there are a such lines, therefore the value of the whole is 6 taken a times, ora, Again, write 1 as many times as there are units in a, and make as many such lines as there are units in 6. In this case the value of the whole will be ba, But the one of these sets of units is plainly equal in number to the other, whence a6 = ba. E Just in the same way we may con- clude, that abc is equal to bac. For write cin the same line 6 times, and make a such lines. The value of each line is 6c, and the value of the whole is abe. 8 Cte +ers c+c+e ro e+e+ete et+tcte ‘§ ctete+re c+et+e Ef CTG+4 “a Again, write c in each line a times, and make &-such lines. The value of the whole, which is plainly the same as before, is now bac. So, since be is equal to cb, abc is equal toacb. From all this it follows, that if a quantity be the product of any number of factors, the order in which thesé factors succeed one another may be altered in any way without changing the value of the quantity. : _ 24. 4=22= 2x 2, and 8 = 23hap 2x 2x 2[art.8]; therefore4 x 8! 2x2x2x 2x 2, or 25. We findd and 4 x 8 to be each 32. So a3 = aaa and a2? = aa, whence a3.a? = aaa.aa@ or a’, In the same way we may finc that the product of any two powers of the same quantity is that quantity raised to the power expressed by the , Sum of the exponents in the factors ARITHMETIC AND ALGEBRA, . farts. 7,8]. We state this algebraically, by writing Ch. aie g* t?, where m and m may be any numbers whatever. For the same reason a”™.a”.ar= Gere Ps and again, a™ ,.a™ .q"™= Meer as te or (a™)s = qs™, In the same way we may find any other power of a power of a quantity. he general expression will be (a iy n ey LF: _ For instance, (32)3 = 36; and, since 32 = 9, it follows that 93 = 36, . We ind that both are 729. 25. By applying what has been laid lown, the reader will have no difficulty n multiplying together any quantities onsisting of one term. The coefficient f the product will be the product of the efficients of the factors, and factors hat are the same, or only differ tn their. xponents, will be collected into one by dding their exponents, observing that hen a factor has no exponent unity is 9 be supplied. For examples, fe to b= 3 x hab = 15 ab, ‘abc.5bd.a"f= 7X 5.aa"bb? fed ) =35a"t1b3edf, | as" bh. (a™ 2 (63)3 — ga” g2™ hbo i = qamtsn p00 { 26. We now come to the multiplica- bn of quantities that consist of more tan one term. a(6 + c¢) means that ‘+ cisto betakenatimes. If we take | (b+ c)+ (b+ c)+ &e. times, and then add them all together, ; shall have a times 6, and a times ¢; ‘e whole sum will be ab + ae. ‘herefore . a(b+c)=ab+ae. ', by taking / (6 —c)+(b—c)+ &e. ames, and adding them all together, + find that . a(b—c)=ab—ac. — Again, regarding a + 6 as a single antity, + b)(e+ d)=(at b)ce+(a+ b)d, by art. [23] ; + b)(c+ d)=c(at b)+d(a+t b), or finally (at b)(e+d)=act+be+adtodd. In a similar manner we find that (a+ b)(e—d) =act be-—ad—bd, and that ‘ : (a— 6) (e+ d) = ac—bcec+ad—bd, Lastly, let us find the product (a —b) (c —d). Regarding a — 6 as a single quantity, we have, as before, (a —6)(¢ —d) = (a—b)c—(a—b)d. That is, from (a — 6)c we are to sub- tract(a — 6) d; orfromac — bcweare to subtract ad— bd. But by art.[13] this difference is ac—bc—ad+ bd. Therefore (a —b)(e— d) =ag¢—bc—ad+ bd. It is easy to see that these principles apply equally when the number of terms in one or both of the factors is more than two. From the results already given we may therefore deduce the fol- lowing general rule: Multiply every term in the one factor by every term in the other, and set down the quantities so obtained, every one with its proper sign for the terms of the product. ‘Fo determine what this sign is, ubserve that the product of two positive, or of two negative quantities is positive ; and that the product of a positive and a negative, or a negative and a positive quantity is negative; or that like signs give a posi- tive, unlike a negative product. 27. The reasons of the part of this rule that determines the signs of the terms of the product will be more ap- parent, if we consider, that to multiply by a positive quantity is to add the quantity multiplied so many times, and to multiply by a negative, to subtract it so many times. Hence the product of a positive quantity by a positive, is a positive quantity added a certain num- ber of times, and is positive. The pro- ‘duct of a positive quantity by a negative, or of a negative quantity by a positive, is a positive quantity subtracted, or a negative quantity added, a certain num- ber of times, and is therefore negative. Lastly, the product of a negative quan- tity by a negative, is a negative quantity subtracted a certain number of times, and is therefore positive, by art. [18]. An example may aid the reader in apprehending that ( — d) x ( — 8) is equal to + d4. A person buys a yards of cloth at ¢ shillings a yard, he keeps 8 b of the yards and sells the rest at d shillings a yard less than he gave for it; how much did he receive for what he sold? He bought a yards and he keeps 0; therefore he sells a — byards, for which he gets c — d shillings a yard. For all that he sells, then, he gets (a — b)(c—d)=ac—ad— bc+ bd shillings, by the rule. Suppose a to be 12, 0 to be 3, c to be ARITHMETIC AND ALGEBRA, 10, andd tobe 2; then ac—ad—bc+t bd = 120 — 24 — 30 + 6 = 72 shillings. Again, since he bought 12 yards and kept 3, he sold 9; and since he paid 10s., and sold for 2s. less, he sold at 8s. ayard. Therefore he received 8 x 9, or 79s., which is of course the same result as before, 28. The following are examples of multiplication ; Multiply By a+3ce—a laws @ 2e+6ac—2ad—ad—3dc+ and collecting the terms, the product is 2a2+6ac—383ad—3cd-+ a, O+a¢++Oe+et+ae—-r7t—-YP-YV-—-Lr—1 Multiply a+za By a—2 C+rar—-ar— ve or a — x, Multiply at+a3+a%°+ar+1 By x—1 or we 1, 29. When there are many factors the product of them all is called theis continued product. Thus (2 + a) (@ + 6) (@ + c)(% + a), &e, is the continued product of these factors, Ve+(atb+oaevrt+(ab+actboxrt+abdbe x +a xt+b a+(at bdxt+abdb “ thie git G rh a, at a + ab + 6 eel 4 ad ad be a sa cd and so on. ‘This continued product forms itself very regularly according to a law which it is not difficult to per- ceive. We shall be able to make more than one important use of it as we go on 30. Since the product of every pair of negative factors is positive, whenever the number of negative factors in a product ig an-eyen number, that pro- | | We find it thus; : z | ! : abe abd, aco d bed +abed duct is positive. On the other hant whenever the numher of negative fat tors in a product is an odd numbe that product is negative; for if we tal, the product, leaving out one negatr factor, there will be an even number | negative factors, and their productis p sitive ; and multiplying this by the neg), tive factor omitted, it becomes negatly It follows from this, that * | ARITHMETIC AND ALGEBRA. 9 (— a)" = a ; for 2 2 isalways aneven number, Simi- larly, (— antl ——res anti ; for 2 + 1is.always an odd number. Thus, (—3)* = (—3).(—3).(—3).(—3) = 81, and (— 3% =(—3).(— 3).(— 3) = — 27. 31. With respect to the multiplication of numbers: the product of any two of the first nine numbers is contained in the following table, so well known by the name of the Multiplication Table. Liz) 3° 4 ohh Ger Bieeih 9 2| 4] 6| 8|10|12)14/16/18° 3/6 | 9 |12| 15] 18] 21.| 24 | 27, 4} 8 |12| 16 | 20| 24] 28 | 32] 36 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 6,12] 1 24 | 30 | 36 | 42 | 48 | 54 7| 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72] 81 This table is said to have been first used by Pythagoras, the famous Grecian philosopher, who lived about 500 years before Christ. It is formed in the first instance by addition; thus, 4 + 4, or 8, is2 x 4; 8+ 4, or 12,1s 3 X 4; and so on, 32. To multiply any number by 10, we add a zero to its right. Thus, since 732 = 700 + 30 + 2, we must have by art. [26] 10 x 732 =10 x 700 + 10 x 304+ 10X2., But by our notation [art. 19] 10x 2=20, 10 x 30 = 300, and so on. Therefore 10 x 732 = 7000 + 300 + 20 = 7320. Similarly, to multiply any number by 100, or 1000, is to add two, or three digits, respectively, to its.right. To find such a product as 9 x 80, observe that it is the same as 9x 8x10. [art. 23]. But by the table 9 x 8 = 72. Therefore 9 x 80 is the same as 72 x10, or 720. Similarly, 2 x 8000 = 16000. 33. By means of the multiplication table, and these properties, we find the product of any two numbers whatever. When the multiplier has only one digit the rule is this: Write the multiplier under the units’ digit of the multipli- cand ; find in the table the product of the units’ digtt of the multiplicand by the multiplier; write the units’ digtt of this product immediately under the multiplier ; find the product of the next digit of the multiplicand by the multi- plrer, add to rt the tens’ digit of the for- mer product, write the units’ digit of the number thus found to the left of the digit last set down, and proceed as before, The reason of this rule will appear by one example. To multiply 9271 by 7. That*iss Multiply 9000:--* 200. -- F0i-tcE 9271 By 7 7 63000 + 1400 + 490 + 7 64897 Here we find the several products from the table as in art. [32], their sum by art. [26] is the product sought ; and when we collect them into one number, we do what is equivalent to the carrying directed by the rule. When both numbers have more than one digit the rule is as follows: Write the multiplier under the multiplicand, units under units, tens under tens, &c. Write down the product of the multi- plicand by the units’ digit of the multi- plier, observing that its units’ digit shall be under the units digit of the mul- tiplier. In like manner, write down the product of the multiplicand by the tens digit, observing that its units’ digit shall be under the tens’ digit of the multiplier ; proceed thus, and, when all the partial products are set down, add them up as they stand, for the whole product. As before, an example will explain this rule. Let it be to multiply 4786 by 2783. That is, to take 4786, 2783 times, and add them all together; or to take it 2000 times, 700 times, 80 times, and 3 times, and add the sums together, or to multiply it by 2000, by 700, by 80, and by 3, and add the products to- gether. By art. [32] and the first rule we have 4786 4786 27383 2783 — 14358 = 3X 4786 14358 38288 /0= 80x4786 38288 33502 /00= 7004786 33502 9572 / 000=2000 x 4786 9572 13319438 =2783x4786 13319438 10 Ifthe zeros be cut off in the detailed operation it will stand as the rule directs. Of Division, _ 34, We have already stated [10], that to divide a by 6 is to find a third quan- tity such that the product of it by 6 shall be a This third quantity it was agreed to represent by :. It follows, then, that a bx a 35. When the divisor is a factor of the dividend, the quotient is simply the dividend with that factor struck out of it. If the dividend be 20, that is 4.5, and the divisor 4, the quotient is5. So if the dividend be a 8, and the divisor 3, the quotient is simply a; for a multi- plied by 8, the divisor, produces a b, the dividend. So a’—a? = a, and aq” — a” — Geen, fora”"-" xa" =a™.[art.24], Forexample, since 243 = 35,and 27 = 3; then 243 — 27 must be 35-3 = 32, or 9, 36. Though the divisor itself be not a factor of the dividend it will often happen that it has some factor that is _ also a factor in the dividend. This factor may be struck out of both of them without affecting the quotient. If a ARITHMETIC AND ALGEBRA, be the dividend, and ac the divisor, 2 ~ < will be the quotient. Forac. ; =ab by [34]; that is, : multiplied by the divisor produces the dividend. Simi- larly, the quotient of 6a ay+sabzis 2 say: — 4, striking out of each the factor 3 ax. 37. When the dividend has more terms than one, the divisor still remain- ing a quantity of one term, the quotient. is found by dividing each term of the dividend by the divisor, in the manner just laiddown. Thus, if the dividend be a+ 6+ ¢, and the divisor d, the quotient will be : a c dt ata because if we multiply this quantity by d the product will be the dividend a+ob-+c. [art. 26]. 38. With respect to the sign of the quotient, we have the same rule as in multiplication, namely, that when the divisor and dividend have like signs, the quotient is positive, when unlike, it is negative. ‘This is deduced from the rule for the signs in multiplication [26], thus: + a—(+ b) = +=, because + b x (+-) = + @; b b y Weed Ayes : +Fa-(—bd= — 7 because — bx & 4 = +a; —a>(+b)= —=,b +bx (-=) = (CF 0) = —, because x PP hh Aire ad : ee ote rod ea as — a> (— 6) = +5, because — b x (+4) =—a@ 39. After what has been laid down we shall have no difficulty in finding that ° 12 a 66 c’ — 16 ac? 64 = kta ; 403 * 3 (25 Poy) isey— 24, Ba oh BO 27 x8 (9 v8 — 3 a? x) v2 = 27 + 3 a(8a2 — ot) 2 322 — a? 40. When the dividend and divisor are both quantities containing more than one term, the operation of division becomes somewhat more complicated. We shall find it better to explain first the reason of the common rule for di- viding one number by another. The rule in algebra depends on the same principles. The arithmetical rule is this: Write the divisor to the left of the chividend ; from the left of the dividend mark off the smallest number of digits that make a number not less than the divisor; find by trials the greatest number of times that the divisor ts contained in this period, and write the ' ; ARITHMETIC AND ALGEBRA. result as the left hand digit of the quo- tient ; multiply the divisor by this digit, and subtract the product from the period marked off; bring down to the right of the remainder the next digit in the dividend, and proceed with the number thus formed as with the first period, writing the result as the second digit in the quotient ; proceed in the same way to find the other digits of the quotient. Take as an example to divide 3978 by 17. We. seek the number which when multiplied by 17 gives 3978 for product. Now, we observe that the “number must be between 200 and 300, for 17 x 300, or 5100, is greater, and 17 x 200, or 3400, is less than 3978. The quotient then is greater than 200, Divisor. Dividend. ) 3978 ( 200 + 30 + 4, or 234. Subtract 17 x 200 = 34|00 ———- 17 57|8 Subtract 17 x 30= 51/0 68 Subtract 17 x 4= 68 —« If the digits cut off by lines be not written, which they need not be, the operation will stand as the rule directs. We find the successive digits in the quotient by guessing at them as well as we can. When we multiply the divisor by the digit guessed, if the product be greater than the partial dividend, the digit is too great; when we subtract, if the remainder be greater than the di- visor, the digit is too small. When the divisor is 12, or under, the operation is much shortened by going through the multiplication and subtraction mentally. This way is what is. called short di- vision. 41. It will, of course, very often hap- -pen; that the dividend does not eontain the divisor any exact number of times ; that is, that there is no exact number which when multiplied by the divisor produces the dividend. For instance, if we had to divide 3985 by 17, we should find, that when 17 x 234 is subtracted, there is 7 remaining. So that 3985 =17 x 234 +7; “here 7 is called the remainder. Now by art. [35] (17 x 234 + 7)-+17.= 234 $a. In general, if a be any number whatever, b.any number less than a, g the exact 11 let us call it 200 + a. Therefore 17 (200 + a), or 17 X 200 +-17.a = 3978, If from each of these equal quantities we take away 17 x 200, or 3400, the remainders will be equal, that is, 17 a= 578, anda= ae fart. 10]. Again, to find a we observe that it must be between 30 and 40; for 17x40, or 680, is greater, and 17 x 30, or 510, is lessthan 578. As before, letabe 30+ 6. Then 17 (30+), or 510 + 17 6=578; and subtracting, as before, 175=68. Finally, we observe that 6 must be 4, since 4 x 17 = 68 exactly. The quotient then is 200 + 30 +4, or 234. We may present this operation thus: Quotient. part of the quotient, and r the remain- der, then Ga gua ts and — coe Bik h Be So that when in dividing any number by another there is a remainder over, it is to be written as divided by the divi- sor, and set down in the quotient. We shall return to these expressions when we come to treat of fractions. 42. A few examples wrought out at length must make the reasons of this rule of division appear very plainly. We now proceed to the division of al- gebraical quantities of more than one term. For instance, to divide 11 a 6? +66+a+ 6abbya+ 6. The course of the operation will show, that, since a stands first in the divisor, it will be most convenient to arrange the terms in the dividend according to the powers of a, putting its higher powers before its lower ones. It will then stand thus, @+6ab+ llab?+ 66°. Now a? is a term in the dividend, and a is one in the divisor; the term a could only come from the multiplication of a? in the quotient by a; therefore a? is a term in the quotient. For the same reason as in the arithmetical example we subtract (a + 0) a, or @ + ab 12 from the dividend, and there remains ; 5a2b+1lab?+ 6 bs. Again, the term 5 a2b in this re- mainder could only come from the multiplication of 5ab in the quotient by a in the divisor; therefore 5 a b is also a term in the quotient. As before, subtract (a + 6)5 ab,or5a2b + 5ab2, and there remains ARITHMETIC AND ALGEBRA. 6 ab? + 6 B8, 6 a 6? could only come from the multi- plication of 6 b2 by a, therefore 6 b?2 is aterm in the quotient ; and subtracting (a + 6)6 b2, or 6ab2 + 6 D3, there re- mains nothing. The quotient then con. sists of the three terms we have found, and is a@+5ab-+ 6 b, The operation stands thus : at b)@+ 6a2b+1llavt+6(a+5ab +682 Subtract (a+ bhat=a@+ atb Subtract (a+ b)5ab= 5ab+ 1llabe 5ab+ 5abe 6ab2 + 6 3 Subtract (a+ 6)6 b= 6ab?2 + 6 b3 The general rule is this: Arrange the terms of the dividend and of the divisor according to the powers of some one letter ; divide the first term in the divi- dend by the first in the divisor, and write the result as the first in the quo- tient ; multiply the divisor by this term, and subtract the product from the di- vidend ; proceed to deal with the re- mainder, tf any, in the same way. 43. By applying the same reasoning as in the last example to all the steps of those which follow, the reader will become familiar with the principles on which the rule rests, 208 — 5a 62+ 2 63)4 aS — 25 a2 bt + 20a65—406(2a+ 5a? — 263 Subtr. divis. x 2a2=4 a5 — 10. a4 b? + 4 3g; Subtr. divis. x 5 a6? = 10 at be 10 at 6B — 4 a2 6 — 25 a2 b4 + 20065 — 4 U6 — 25 a2 64+ 10 ahd —406+10a65_466 Subtr. divis.) x (— 2:68) = 4p 63+ 10ab5 — 4 66 s 44, a—l)@—1l(@t+at+l OF — G2 a? — 1 ea a—] a—I1 Examining this last example, it is plain that in like manner at — 1, or, generally, that a” — 1 may be divided by a — 1, without leaving ‘any remain- der, and that we should have a" — 1 a—l But we should find that a? + 1l,a*+ +1, or generally a” + 1 cannot be divided by a —1 without leaving a remainder, 45. In like manner, if we divide a + 1, or a + 1, or any similar quan- tity where the index of a is an odd number, by a+ 1, we shall find that there is no remainder over ; but if we try to divide at + 1, or any similar ex- pression where the index of a is an even number, by a+ 1, there will al- =a"1+ a-2+ &.+at+ 1, ways be a remainder over. Now if 2 stand for any number, 2 2 +1 will stand for any odd number ; for every odd num- ber is twice some other number with 1 added to it. a®"+1+ 1 then can always be divided exactly by a + 1, and we find Deer | Ss om = @"— a" 4+ &. +a2—atl. Once more, a2 — 1, at — 1, and all similar expressions in which the index of a is an even number, can be divided by a+ 1 without leaving any remainder; but a? ~— 1, a — 1, &c., where the in- dex of a is an odd number, always leave a remainder. Now 27 is the general representative of an-even num- ber, and therefore a?" — 1 can always be divided without remainder bya +1, We shall find a" — 1 a+il = a") g-2 4 &e, - a tae 1. ARITHMETIC AND ALGEBRA. 46. As a last example, let us propose to divide 1 by 1+ 2. wit xr)l (—xv2+ #2 — 23+ &e, ££ rae — & —xr— x , ve Y ali wae a — 7 The division, it is plain, will never come to an end, and we have ] Be Ex 47. Of course all these results, as they are true for all quantities, are also true for numbers. Thus, in the ex- = l— e+ 2 — xe+ xt — &e. ; qer +} +] pression for ate [art. 45] make nm = 2, then 5 oi =at—@+t+a—atl. Now, let a = 4, then 45+] * =4t— 48+ 444+ 1, eae “4 “+e or £2024--+- T 4+1 that is, = 256 —64+ 16—4+1, 1025 —_ 2056 5 We said that a® +1 cannot be di- vided by a —1 without leaving a re- mainder. Yet make a= 3 andv=4, and we find that 34 + 1, or: 82, is di- visible without remainder by 3 — 1 or 2. The reason of this seeming discrepancy issimple. If we dividea+ + lbya —1, we find for quotient a a oe - P- oi 8 ale so that our assertion holds true in ge- neral; but when 3 is put for a, it hap- 2 pens that becomes — or T, So 3 — 1 that the remainder, as a distinct-part of the quotient, is lost. _ In the result in art. [46], let us put — 2 for x; then that expression be- somes ——— ap = 1A (-2) 4 (- 28 (= 29+ &e. or by art. [30] fee b=l+2+4+8+16+-&c. which is seemingly false.. But if we go sack to the operation by which we ar- rived at the expression in art. [46], we | j 13 shall find that there was always a re- mainder over, which we included in the &c., and that the true way of writing the result would have been 1 1 a ae Lita or J 1 i v2 ——_—_——— — HI ———— 1+2 lta or : i xt ——_— =1-7+a2— 23 : PpP aL Tie lp ae and so on. Now if we take any of these expressions, for example the last, we shall find it true ; because, putting — 2 for x, it becomes wh Lt. bh Si 16. Once more, if for x in art. [46] 2 be put, the result becomes = Hl-1+1-1+1-1+41-&e. This is a result which has puzzled some eminent mathematicians, who thought that the &c. must comprise a set of terms exactly the same as those that go before it, and could not understand how a series of numbers, which, when added together, is alternately 0 or 1, according as an even or an odd number of them is taken, an ever be equal to > it the remainder be taken into the ac- count, the difficulty disappears, for we have 1 i => 2£ Ite 1+2’ or making v =1, 1 by 1 ie Pak 1 xt Se ee ay ea 2? l+a I + a" or making @ = 1. 1 1 and so on. These remarks are introduced to show how careful we must be when we use algebraical results, not to forget the steps by which we arrived at them. Of Whole Numbers. 48. Such numbers as 2, 5, 9, 12, 90, &c.are called whole numbers, or integers, (a Latin word meaning whole, ) in distinc- tion to such numbers as 53, 3, &c, 14 49, When a whole number is the pro- duct of any other whole numbers, any one of them is said to measure it; meaning that the number can be divided by that iactor without leaving any re- mainder. Thus, 20 is the product of 2 and 10, or of 4 and 5; therefore 2, 4, 5, and 10 measure 20. The reason of this term is plain. We can measure 20 gallons by means of a vessel that we know to contain 2 gallons, or by means of vessels containing 4, 5, or10 gallons ; but we cannot measure 21 gallons by any of these, nor 20 gallons by a vessel containing 6. 50. A number that is measured by any other is called a muléiple of that other; thus, 20 is a multiple of 2, of 4, of 5, or of 10. The numbers 2, 4, 5, and 10, again, are called swbmudtiples (that is, under multiples) of 20. 51. Leta = q 8, thencawill be equal to cq 6; for if equal quantities be mul- tiplied by the same quantity, the two products must be equal. ‘Therefore 6 measures caas well as a; so that if one quantity measures another, it measures any multiple of that other. For ex- ample, 5 measures 15, therefore it mea- sures 6 X15, or 90. On the other hand, if one number be measured by another, it is measured by all the factors of that other. 52. Every number is the product of itself by unity; thus, 19 is the product 1 X19. When a number has no other whole factors but itself and unity, it is called a prime number. Thus, 1, 2,3, 5, 7,29, 31,101, &c. are all prime numbers. 53. If we wish to discover whether any number is a prime, the only way we can do it is by trying if we can find some number that will measure it ; if we are sure that none of the numbers less than itself measures it, we are of course sure that it isa prime. Now, if it be measured by any number that is not a prime we have seen that it must also be measured by all the factors of that number [51], and some of these factors must be primes. If the number in question, then, be not measured by any prime number less than itself, it is not measured by any other number, and it is therefore a prime. Again, let a be the number in ques- tion, and let 6 be its square root [11]. If a be measured by any number greater than b, it must also be measured by some number less than 0; for since b multiplied by b produces a, the number by which any number. greater than ARITHMETIC AND ALGEBRA, b must be multiplied so as to produce a, mustbe less than 6. From all this it follows, that if any number be not measured by any of the primes that are not greater than its square root, it isnot measured by any number what- ever, and is consequently itself a prime. To determine that 47 is a prime, for in- stance, it is sufficient to be certain that it isnot measured by 2, 3, or 5; the only three primes not greater than its square root, which is between 6 and 7. So 167 is a prime, for it is not measured by 2, 3, 5, 7, or 11, the only. primes not greater than its square root, which is between 12 and 13. 54. Two numbers are said to be prime to each other when there is no number but unity that measures both of them; thus, 35 and 12 are prime to each other, though neither of them is itself a prime. On the other hand, if there be two numbers, both ef which are measured by a third number other than unity, they are said to have a com- mon measure. 15 and 25 have a com- mon. measure 5; 369 and 270 have common measures, 90, 45, 30, 18, 15, 10,;.9, 6, 5; 3; and/2: : 55. Let a= bc and a = b'c; hete a and a’* have a common measure, which is c. Now atad=c(b+D’, and a—d=c(U=— bd); so that c measures a + a’ and a — @, If two numbers, then, have a common measure, it also measures their sum and difference. Thus, 63 and 85 have 4 common measure 7; 7 also measures their sum 98, and their difference 28. 56. If any two numbers a and bd have a common measure c, and if on dividing aby 6 there be a remainder r, ¢ will measure r also. For if q be the whole part of the quotient, then by[41] a. 27 ONE If from each of these equal quantities q 6 be taken away, the remainders will be equal, and therefore _ . a—gqb=r. Now c measures a; it also measures b it therefore measures q 6 [51]; anc therefore it measures the difference 0: those quantities [55]; and that differenet is 7 Thus, 3 measures 57 and 12 _* The notation in Algebra is often made muel simpler by denoting ditferent quantities, not by dif ferent letters, but by. the same letter ditferentl marked, as a, a’, a, &¢, OF A. dy, dg, SC ARITHMETIC AND ALGEBRA. 15 dividing 57 by 12, there is 9 over, which common measure of two numbers. Let is also measured by 3. 57. On these principles we may esta- blish the rule for finding the greatest Divide a by 6 and let 7 be the remainder, then a Divide 6 by 7 and let 7” be the remainder, then b Divide r by 7’ and let 7" be the remainder, then 7 = Divide 7” by 7” and let there be no remainder, then ¢ We have gone on dividing a by 8, b by the remainder 7, r by the succeeding remainder 7’, &c. till 7”, the third re- mainder, is found to divide 7”, the pre- ceding one, exactly. 7” is the greatest common measure of a and &. In the first place, it isa common mea- sure ofa@and J. Forit measures 7” ; it therefore measures q” 2” [51]; and there- fore q’r’+r’ [55], thatis 7. In the same Way, since it measures 7’ and 7, it mea- sures q’7 + 7”, orb. Finally, since it measures 7 and 6, it measures q b + 7, ‘or a. In the second place, it is the greatest ‘common measure of a and 6. For every common measure of a and 6 measures ry [56 |]; and every common measure of 6 and r measures 2”, and every common measure of rand 7’ measures 7”. Now r’ is itself the greatest number that measures 7” and 7”, and therefore 7” is fhe greatest number that measures aand 8. It is plain, that as the quantities 7, 7’, 1’, &c. always go on diminishing, if the operation does not stop sooner, we shall at last come to a remainder that shall ve unity, When this is the case, the fumbers are prime to each other, for they jave no common measure but unity. | 58. The rule is this: Divide the xreater of the numbers by the smaller; hivide the smaller by the remainder of he last division, vf any ; divide the : i | i] e—axr—anxr? + x) at — at ; a—@r—@ar*+ax the numbers be a and 8, of which 8 is less than a. | Il q b a ¢ ; Gr Ar. = g's yr”, Jirst remainder by the remainder of the second division, of any ; proceed in this tell some remainder divide the preceding one, that remainder is the greatest com- mon measure if it be unity, the num- bers are prime to each other. Of this rule take as an example the numbers 234 and 3348. + #, et 234)3348(14 234 _ 1008 936 Ist remainder 72)234(3 216 2nd remainder 18)72(4 72 Here 18 is the greatest common mea.’ sure sought. In the same way, it will be found that 47 is the greatest common measure of 2961 and 799, and that 824 and 319 are prime to each other. 59. The rule for finding the greatest common measure of two algebraical expressions isthe same. For example, to find the greatest common measure of qr = a; . and eo—Aexr—ax + x3, Dividing the expression in which the highest power of a occurs, by the other, we have (@+ Zz pig exer+a«¢—arv— at a@ar— ata —axz — a Ist remainder 2 a? a? 3efore dividing by this remainder, we ybserve that it is measured by 2 22, vhich does not measure the first divisor, he quantity to be divided. 2 a? is nerefore not a factor in the common qeasure of the remainder and _ first ivisor, and therefore cannot be a factor 1 the common measure we are seeking 57], so that it may be struck out of ur new divisor without affecting that ommon measure. 7—2)8—AxXx—avr+e(a—ax | a — ax | age" + x — anv — 82V SOR: We find that a? — x? divides the former divisor without remainder, it therefore is the greatest common measure sought. This operation for finding the common measure of algebraical expressions is scarcely ever used in practice. 60. If a and b be any two numbers, and ¢ any prime number that neither measures a nor 0, then c does not measure their product a b.. Thus 3 measures neither 5 nor 8, so it cannot measure 40, thei product. This may be proved as follows. . First, let 6 be less than ce, and suppose, for the sake of argument, that @0 is 16 measured by c. We have by [41] Ce gt ov and if we multiply each of these equal quantities by a, the products must be equal; therefore ac=qab-+ar. Now ac and qab are both measured by c, and therefore as in [56] ar 1s measured by c. Again, let CHayrt+y, then as before ac=aqdrt+ar, and as before ar’ is measured by c; and, similarly, if C— og + pe we should prove that a7” is measured by c. Now the numbers 7, 7’, 7”, &c. always go on diminishing, and at last some one of them must be unity, since c is a prime, and therefore cannot be mea- sured by any number but unity. But on the supposition that a b is measured by c, we have shown that a7, a7’, &e, must also be measured by c. It would follow, then, that 1.a@ can be measured by c, which is contrary to our first sup- position. So that it is impossible that ec can measure a0, unless it measure a at the same time. Next, if b be greater than c we have C= ee, where 7 is less than ¢, and multiplying by a this becomes ab=Gac+ar. Now, if ab be measured by c, a7 must be so too, which we have just proved to be impossible. 61. It follows from this, that if we take any number, and find a set of prime numbers which when multiplied together produce it, that is the only set of prime factors the number can have. For if 2 be a number the product of a, b, and c all primes, no other prime d can measure it; for d cannot, as we have just seen, measure a b, and there- fore it cannot measure abe, that is 7. No number, for instance, which is the product of 2, 5, and 7, or of any powers of these numbers, can be measured by 3. So the product of any set of primes is prime to the product of any other set, all of which are different from the first. For no prime that measures the one pro- ARITHMETIC AND ALGEBRA. duct can measure the other, and there- fore no number not a prime can mea- sure both. _ 62. Every number then can be reduced into only one set of prime factors. The way in which we so reduce it, is by dividing it continually by all the prime numbers that will measure it, till it be brought down to a number which we find to be a prime. Take, for example, the number 8316 ; we divide by 2, which we can do twice; then by 3, which we can do three times ; it cannot be divided by 5; and when it is divided by 7, the quotient is 11, a prime. The operatior stands thus : 2)8316 2)4158 3)2079 3)693 3)231 7)77 1] — So. that 8316 = 2%. 324 70k gee 360.= 28..°33,.5,and 210 — 25 3 pas 63. It follows from [61] that all th numbers that measure any number, mus necessarily be the products of some 0 the prime factors of that number. N¢ number can measure 210, but1, 2,3,5,7 or the products of some of these num bers. ‘The common measures of tw numbers must be the prime factors the have in common, and the products 0 these prime factors. Thus, 420 = 2? 3.5.7, and 360 = 28. 32.5. The fag tors they have in common are 2?, 3, ani 5, so that their common measures €a’ have no factors but these. Their greates common measure is plainly 2?.3 .: or 60. Their other common measure are 2, 3, 4, 5) 6, 10,°12,95,920, 303mm one number have a prime factor whic is not in another, it may be struck ov of the first without affecting their com mon measure. | 64. If a,b, and c be three number; and m be the greatest commun measul of a and b, the greatest common mez sure of a, b, and ¢ will be found by fine ing the greatest common measure ( m and c. For, by the last article, : and its factors are the only numbe’ that measurea and b; and the greate. of these numbers that measures c¢, th’ is, the greatest common measure of | and ¢, will therefore be the greatest th. measures a, b, andc. If the numbers |. 1512, 588, and 330, the greatest con mon measure of the two first 1s 84; al ARITHMETIC AND ALGEBRA. the greatest common measure of 84 and 330 is 6. Therefore 6 is the greatest common measure of the three. So, if there be a fourth number, the greatest common measure of the four is found by finding the greatest number that mea- sures at once the fourth, and the greatest common measure of the three. 65. A common multiple of two num- bers is a number which both of them measure. Thus, 90 is a common mul- tiple of 6 and 15. It is useful to know that the least common multiple of two numbers is found by dividing their pro- duct by their greatest common mea- sure. The greatest common measure of 210 and 360 is 30, and therefore their é . 210X360 least common multiple is Ware 2520. The reason of this appears when we consider that 210 = 2.3.5.7, and 360 = 23.32, 53 so that no number less than 2%. 32.5.7 (that is, 2520) can be measured by bothof them. But 23. 32. 5.7 is their product with 2.3.5 struck out of it, that is, their product divided by their greatest common measure. The least common multiple of three or more quantities may, in like man- ner, be found, by resolving them into 67. Again, 17 their prime factors, and considering these as in the instance above. Thus, the least common multiple of 144, 210, and 360 is 5040; for 144 = 24. 32, 210=2.3.5.7,-and 360/= 29 73725, so that no number less than 24. 32.5.7 (that is, 5040) can be measured by all of them. 66. We now proceed to another way of considering numbers that leads to important consequences. Let us take any number, as 98237, we may write it in this way, [art. 19 j 7 +30 + 200 + 8,000 + 90,000, that is, 743 10+2.102+8,109+ 9,104, Now 3.10, 2.162, 8 . 103, &e. are all measured by 2 and by 5, and therefore if the term farthest to the left, that is, if the units’ digit be divided by 2 or by 5, the remainder must be the same as when the whole number is divided by 2 or by 5. It is only when the units’ digit of the number is 0, or a multiple of 2, that this remainder, on dividing by 2, is 0, and therefore it is only in these cases that the number is measured by 2. So it is only when its units’ digit is 0 or 5, that a number is measured by 5. 3.10=3.(10 -1+1)=3.00—1) 4+ 3, 2.102? = 2.(102 ~1+ 1) =2.(102— 1) + 2, 8.103=8.(10?3—~1+1) =8.(10? —1) +8, and so on. Therefore the number may be written 7+3(10—-1)+34+2.(102—1)+2+8.(0108 —1)+84+9,(10*—1) +9, or writing the same terms in a different order 74+34+2+84+94+3.(00—1)+ 2 (10?—1) + 8(10?—1)+9.(104 — 1). Now by art. [44] all the terms 3.(10—1), 2 (102 — 1), 8.(103 — 1), &c. are mea- sured by 10 — 1, that is, by 9. Sothat the remainder when 7+3+2+8+9 is measured by 9, is the same as the remainder when the original number is measured by 9. The same principles apply to all numbers, and therefore the remainder when any number is divided by 9 is the same as when the sum of its digits is divided by 9. In like manner, since every number measured by 9 is also measured by 3, the quantities 3.(10 — 1), 2.(16? — 1), &e. are all measured by 3. It follows, as before, that when the sum of the digits of any number is divided by 3, the remainder is the same as when the number itself is divided by 3. As examples of these properties, 17 is the sum of the digits of 278; dividmg 17 by 9 the remainder is 8, and dividing it by 3 the remainder is 2, therefore when 278 is divided by 9 or 3, the remain- ders are respectively 8 or 2. The sum of the digits of 1287 is 18, which is measured by 9, and therefore 1287 is also measured by 9. 68. The common way of verifying the multiplication of two numbers, or prov- ing it to be correct, by casting out the nines, is founded on the property demon- strated in the last article. It is this: Take the sums of the digits of the mul- tipiier, of the multiplicand, and of their supposed product separately; write down severally the remainders when these three sums are divided by 9; take the product of the first and second of these remainders, the remainder when this product is divided by 9 ought to be the same as the third. For let N and N’ be the two numbers; when N is divided by 9 let the remainder be 7, and Cc 18 der be 7’. Then by [41] N =.9.9 +75 N= 9q'+ 7’. Multiplying these equal quantities to- gether, the products must be equal, therefore NN’=8lqgq¢+9qrt+9qrtrr, and as 81qq', 9q/7, and 9q7’, are all measured by 9, the remainder when 77” is divided by 9,\is the same as when N N’ is divided by 9. Now, by art. [67] when we divide the sum of the digits in N by 9 the remain- der is7, and when we divide the sum of the digits in N’ by 9, the remainder is 7’; and, from what precedes, when we divide the sum of the digits m N N’ by 9, the remainder ought to be the same as when 77” is divided by 9. If it be not so, we are sure that the multipli- cation NN’ has not been rightly per- formed. Take the example of multipli- cation in [33] as an instance. Sum of dig. of 4786 = 25. Rem. = 7. Sum of dig. of 2783 = 20. Rem. = 2. Sum of dig. of 13319438=32. Rem.=5. 70. Again, observe that ARITHMETIC AND ALGEBRA. when N/ is divided by 9 let the remain- The last remainder is the same as when 2x 71s divided by 9. This proof is not quite perfect, for though it always holds when the mul- tiplication is right, it may sometimes hold when it is wrong. Ifthe product be too great or too small by any mul- tiple of 9, the remainder when it is divided by 9 is the same as if it were right; and therefore the proof will not in such a case show it to be wrong. 69. We found 98237 =7+3+2+849 + 3(10 —1) + 2(10? — 1) + &e. From each of these equal quantities take away 7+3+24+8+9, or 29, and the remainders will be equal. Therefore 98237 — 29 =3(10 —1) + 2(102 — 1) + &e. We have seen that the second expres- sion is measured by 9, and therefore 98237 — 29, or 98208 is measured by 9. In general, if from any number the sum of its digits be subtracted the re- mainder is measured by 9. 3.10 =3(10 +1—1)=3.(104+1)—3, 2.102= 2(102 —1 + 1) =2.(102-1) + 2, 8.108 = 8(10? + 1 — 1) ='8.(10 + 1) —8, 9.104= 9(10¢—-1+1)=9.(104+—1) +9; so that the number 98237 may be written 7—-34+2-—849 + 3(10+1) + 2 (10? —1) + 8 (108 + 1) + 9 (104 — 1), Now by [45] all the quantities 10 + 1, 102 — 1, 10? + 1, and 104 —.1,. are mea- sured by 10 + 1, thatis, by 11. There- fore if 7 —3 + 2—8 + 9 be divided by 11, the remainder is the same as if the original number were divided by 11, When this remainder is 0, the number is measured by 11. So that if the digits of any number be taken alternately, and the sum of one of the sets be subtracted from that of the other, when the differ- ence is 0, or 11, or any multiple of 11, the number is measured by 11, Observe that 98237—(7 —3 +2~8 +9) must be divisible by 11. In the same way, generally, if N be any number, A the sum of its alternate digits beginning with the units’ place, and B the sum of the other digits, the expression N — (A —B) is measured by 11. 71. This way of expressing numbers would lead to many other properties of the same sort. We shall give two more. Take the number 89764; it may be written 64 + 9700 + 80,600, or 64 + 97.100 + 8.1002. Now 100 and all its powers are mea- sured by 4, and therefore since 4 mea- sures 64, it measures 89764 also. In general, whenever 4 measures the two last digits of a number, it measures the number itself. Similarly, if the three last digits of a number be measured by 8, or the four last by 16, the number is measured by 8 or by 16. In the same way, if the two last digits be measured by 25, the number is measured by 25. If the three last be measured by 125, the number is measured by 125, and $0 On, | ARITHMETIC AND ALGEBRA. 19 72. Take the number 8,937,524,361; it may be written 361 + 524.1000'+ 937 . 10002 + 8.10003, or, as in[70], 361 — 524 + 937 — 8 + 524 (1000 + 1) + 937 (1000% — 1) + 8 (10003 + 1). As before, 1000 +1, 10062—1, 10003+1, are all measured by 1001, and therefore by all the factors of 1001, and therefore by 7, for 7 x 143 = 1001. So that the number proposed divided by 7 leaves the same remainder as 361 — 524 + 937 —8does. If, therefore, we take any number and divide it into periods of three digits each beginning from the right, and then take the difference of the sums of the alternate periods, when that difference is measured ,by 7, the number itselfis. For example, take the number 2,724,016,614,837,540,988, 988 + 837 + 16 +2 = 1843, 540+ 614+ 724 = 1878. The difference of 1878 and 1843 is 35, which shows that the number is mea- sured by 7. In the same way, since 1001 is mea- sured by 13, if the same difference be measured by 13, the number is mea- sured by 13. 73. We have seen[19], that our way of writing numbers consists in an agreement, that the value of every digit shall be ten times as great as if it held the next place towards the right. We owe a very great deal of our present knowledge to this simple invention, which is so admirably adapted to the ends it has to serve. It came fo us from the Arabs about A.D. 1000, and _ was not known to the ancients, though many of them thought and wrote very profoundly about numbers. Instead of agreeing that the digits should increase invalue ten times, it might have been set- ~ tled that they should increase eight, or twelve, or any other number of times. Ten is called the base of our scale of notation, as eight would be the base of a scale where the digits increased eight- fold, or twelve, of one where they in- creased twelve-fold. When we come to consider decimal fractions we shall see reason to think that twelve would have been a more convenient base than ten. Men were, perhaps, led to name the numbers according to a scale proceed- ing by tens, on account of the facility that the ten fingers would then give them in counting; and, afterwards, when they thought of ciphering they would naturally use the same seale, 74. In our seale, or the decimal scale asit is called, we can express any num- ber by means of nine digits, and zero or nothing. | In the seale whose base is eight, we can express all numbers by means of seven digits and zero, and so for others. A number may be transferred from the decimal scale into any other by the fol- lowing rule: Divide the number by the base of the new scale, and write the re- mainder as the units’ digit sought; di- vide the quotient by the base again, and write the remainder as the digit next the units’; proceed in this way till a quotient is obtained less than the base, this quotient is the digit of the highest order in the number in its new form. Whenever there is no remainder, 0 is the corresponding digit. Let it be re= quired, for instance, to present the num- ber 2931 in the scale whose base is 8. Dividing by 8, we find 2931 = 366 x 8 +3. Again, dividing 366 by 8, we find | 2931 = (45 x 8+ 6)8 +38 = 45 xX 8 +6 x 8+ 3, and dividing 45 by 8, 2931 =5.8+5.8+6.84 9 If this be written 5563, and it be under- stood that the second digit is multiplied by 8, the third by 8%, and the fourth by 83, each digit is 8 times as great as if it stood in the next place to the right, and therefore the number is expressed in a scale whose base is 8. So if (10) and (11) be the two addi- tional digits necessary in the seale whose base is 12, the number 13583 transferred into that scale becomes 7 (10) 3 (11), Similarly, 139 transferred into the scale whose base is 2, becomes 10001011,which is equivalent to 27 + 23-+ 2+ 1. Since 54 is equal to 2 . 3°, it will be expressed in the scale whose base is 3 by 2000. 75. To reduce a number from any other scale into the decimal, the rule is this: Multiply the digit farthest to the left by the base of the scale in which the number is expressed, and add the next digit to the product ; multiply the sum again by the base, and add the third digit ; proceed in this way till the units’ digit is added, the result is the number in the decimal scale. For example, if : Ce 20. 3465 be a number expressed in the scale whose base is 9, it is equivalent to 3.93 + 4.92 + 6.9 + 5, aud this is the same as (3.9 +4) 9? + 6.9 + 5, Yr as {(3.9+4)9+6}9+5 and this last expression merely indicates the operations directed by the rule. The working is this. 3465 9 31=9x344 9 285 = 9 x31 +16 9 2570 =9 X 285 +5 In like manner, 19(11)74 ARITHMETIC AND ALGEBRA. 3000, where the base is 4, is 192 in the decimal scale. 76. Some of the properties we have been considering are general, and are the same in every scale of notation; such are those depending on the nature of prime factors. Some, again, are owing to the particular scale in which the number is expressed ; such are those furnishing tests of the capacity of a number to be divided by certain others. In the decimal scale a number is mea- sured by 9, when the sum of its digits is measured by 9; and in the scale whose base is 11, a number would be measured by 10, when the sum of its digits was measured by 10. There is a way of exhibiting the results of this latter in the kind very generally. Let us take the base whose scale is 12, is 37960, and expression N=A,+A,R+ A: R? + A, R? + A, R4 + &c* as in art. [70] it may be put into the forms N=A,+Ai+ Act A, + Aj+ &e. + Ar(R —1) + Ac (R?— 1) + A, (R?— 1) + A,(R4 —1) + &e. N=A,+A,+A,+ &e. —(Ai + A, + &e.) + A(R + 1) + Ay? — 1) +A, (R8-+ 1) + Ay (Rt — 1) + &e. Every number that measures R, mea- sures A, R, A, R%, A, R*. &e. [art 51], and therefore in the first of these expres- sions every factor of R that measures Ag, measures N [art. 55]. Every number that measures R — 1, measures A, (R — 1), Av (R?— 1), &e. [art. 44], and therefore in the second ex- pression every factor of R — 1 that mea- sures A, + A; + Az + &c. measures N. Lastly, every number that measures R +1 measures Ay (R+1), Ao (R?—1), A, (R?+ 1), &e. [art. 45], and therefore in the third expression every factor of +1 that measures A; + A, + A,t+&e. — (A, + A, + &c.) measures N. Now, let us suppose 7 to be the base of the scale of notation in which the number N is expressed, and instead of R in the expressions, let us write 7”. They become N = A, + Aiz® + Aor 2* + Agr + Ayr” + &e. N=A,+ Ait A, + A, + A, + &e. + Ai (7™ — 1) + As (72 — 1) + A; 7" — 1) + A,(r* = 1) +. &e. N=A,+A.+ A, + &e. — (Ai + A; + &e.) + AiG” +1) + A, @"— 12) + As (* +1) + Ay (r#* — 1) + &e. And, as before, N is measured, first, by all the factors of 7” that measure A, ; secondly, by all the factors of 7” —1 that measure A, + A, + As+ &c.; and, thirdly, by all the factors of 7”+1 that measure A, + Az + A, + &e — (Ai + A,+&c.). For instance, let ee 1) and 97 = 2 § then 7” =100, 7” —1=99, and7” +1=101. Also N= A, + Ai 100 + A, 10000 + Az 1000000 + &e. Or we suppose the number N, which is now expressed in the decimal scale, to be divided into periods of. two digits each, of which A, is farthest to the right, A, next toit, and so on. Then by the three last mentioned properties N is measured; first, by such of the numbers 2, 4, 5, 10, 20,25, 50, (the factors of 100,) and 100, as measure A, ; secondly, by such of the numbers 3, 9, 11, 33, (the factors of 99,) and 99, as measure A, + Ai + Az &c. ; and thirdly, by the number 161 (a prime) if it mea- sure * See note to art. 55, ARITHMETIC AND ALGEBRA, A, + AetA, + &e. — (Ai + As + &e.) So if 7 be 8, and 7 still 2; a” = 64,7" —1 = 638, 7” + 1= 65; and the number is supposed to be ex- pressed in the scale whose base is 8. Di- viding it into periods of two digits each, from the right, we have it measured ; first, by such of the numbers 2, 4, 8, 16, 32, and 64 as measure A,; secondly, by such of the numbers 3, 7, 9, 21, and 63 as measure A, + Ai + Az+ &c.; and thirdly, by such of the numbers 5, 13, and 65 as measure A, + Az + &c. — (Ay + Ast &ce.) It is plain, that by varying 7 and 2 we may find as many properties of the same kind as we please. These pro- perties are of little practical use, and the expressions are inserted chiefly as affording examples of the comprehen- siveness of algebraical language, and showing how little beyond an under- standing of the symbols is necessary, to enable us to arrive at results seemingly abstruse and difficult, Of Fractions. 77, It will often happen, when we are expressing the magnitude of any thing by means of a number, that the unit we employ is not contained any exact number of times in the thing in question, but that there is a part over less than the unit. We can still, how~ ever, express the magnitude of this part by means of the same unit. We suppose the unit to be broken down into a stated number of equal parts, and then we say how many of these are contained in the portion that is over. This way of expressing the magnitude of things is called a fraction, from a Latin word meaning to break. We say that a distance is eight miles and two thirds of a mile, meaning, that in addition to eight miles we must divide a mile into three equal parts, and take two of these. 78. Let us consider any fraction, such as nine tenths. Here unity is to be divided into ten equal parts, and of these nine are to be taken. Ten times any one part must make unity ; therefore ten times the nine parts must make nine times unity: so that ten times nine tenths make nine, and therefore nine tenths is the number which, when mul- tiplied by ten, gives nine for product. Now it has been agreed [10] to write 21 that number =, and therefore the frac- tion is to be written in the same way: So any other fraction as two thirds is ‘ 2 to be written es The lower number showing into how many parts unity is to be divided, is called the denominator ; the upper one, showing how many of these parts are to be taken, is called the numerator. On the same principles, whenever t 15 ; there is such a number as 7 resulting from division, it is a fraction, and may be read as such. Its meaning is 15 parts of unity divided into 17 equal arts. Such quantities as — ae eek = a eee are called algebraical fractions. 79, We have already seen [36], that when the same quantity is a factor in the divisor and dividend, it may be struck out of both without affecting the value of the quotient; and so when it is a factor in the numerator and de- nominator of a fraction, it may also be struck out of both. Thus, wo Te ; be b 25 5 and —- = —. Whenever, then, there 65 13 is a common measure of the numerator and denominator of a fraction, it can always be reduced into one of equal value, but of which the numerator and denominator are smaller numbers, or less complicated expressions, Thus, 5005 43316 common measure is 91, and 6a—6 ay + 2ay? —2y% ph OE Peek bE tah eM 12a2@— 15 ay + 3y? 55 can be reduced to aye where the to Gaz? + 2y2 12Qa—3y’ the common measure being a —y. When the numerator and denominator have no common measure, the fraction is in its lowest terms, and is called an irreducible fraction. 80. For the same reason, if the nu~ merator and denominator of a fraction be multiplied by the same quantity, its : 9 yalue is not changed. For instance, rr 22 45 a . v1 and a va By means of this property, when several fractions have different denominators, they can, with- out altering their values, be changed into fractions that shall all have the same denominator. To do this, Mu/ti- ply the numerator and denominator of every one of the fractions by the product of all the denominators, except its own. — — e and—;; ta Eh ® the second bf’ Thus, if the fractions bé —, = as Be adf bdf” , ne OL and the third bap? (3 1 the first becomes értie they ibe cde 45 0 60.60" an and < , they become 48 60° When the denominators of any of the fractions have a common measure, the common denominator resulting from this rule is greater than is ne- cessary. If the least common mul- tiple of all the denominators be found, as directed in art. [65], the fractions may be reduced into others whose com- mon denominator is this multiple. This is done by multiplying the numerator and denominator of every fraction by the common multiple divided by the denominator of the same fraction. Take, for example, the fractions =, ‘3-4 reg The least common multiple of 12,4,and 9,is 36. Multiply the nume- 36 , or 12 : 5 rator and denominator of a by 3 36 4 36 3, of — by —, or 9 — —- 3 a PY ar ,and of — by oa 0 36’ 36 81. Since we may multiply the nume- rator and denominator of a fraction by any quantity without changing its value, that quantity may be — 1. If we mul- tiply the numerator and denominator of a—b b—a ould d—c without altering its value, the signs of all the terms in the numerator of a fraction may be changed, if, at the same or 4, and they become by — 1, it becomes Sothat ARITHMETIC AND ALGEBRA, time, the signs of all the terms in its _ _ denominator be changed. b .at+6b 82. By [37] = aes mph oe wath, & = Bee Toa add or subtract fractions, then, that have the same. de- nominators, we add or subtract their numerators. To add or subtract frac- tions that have different denominators, we must first reduce them to fractions that have the same denominator. Baa: 12 12? and 1 ] a : Fina a Bae Be: es by eRe aw and a Cc S ga Fen vont a: a—3b 5a—-b V7a—-7b C amie ge aro arb a—b a+b a+2ab-+ a@—2ab+ & o a? — & is i a 4ab az — 62 83. When the numerator of a frac- tion is less than the denominator, the fraction is less than unity, and is called a proper fraction. When the denomi- nator is less than the numerator, the fraction is greater than unity, and is : 3. called an wnproper fraction. 7 sa pro- 5 : ; : per, and q @n improper fraction. A, number made up of a whole number and a fraction is called a mixed number ; 1 1 as 13 + 3? oF oy 37h as these are usually written, 134 and 7s. 84, Kvery whole number may be con- sidered as an improper fraction whose denominator is unity, and it may be re- duced to an improper fraction of any other denomination, by multiplying it by the number that is to be the denomi- nator, Thus, 7 is the same as <, or ARITHMETIC AND ALGEBRA. fey. + 84 10’ number may be written as an improper fraction, by changing the whole part of it into a fraction, with the same deno- minator as the fractional part. Thus, In this way any mixed 24 29 = Aes 4 “teas al —. Similarly 6 6 6 b b a+ —= ¢ ota g c An improper fraction may be reduced to a whole or a mixed number, by di- viding out the numerator by the deno- minator. Thus, af = ae 8 8 85. To multiply a fraction by a whole number, multiply its numerator by the a For c X — means whole number. 5 [art. 7] = + 7 + &c. taken c times ; that is, by [82], at+tata+t &e. [taken ¢ times | b CLO 33 30 a. So 10 x SS a eed fi Tag? 86. To multiply by a fraction is to multiply by its numerator, and divide by its denominator. Suppose we have a 2 ie we have multiplied ¢ by a quantity 0 times too great, and therefore the pro- duct is 6 times too great. For the true ac B or the quantity which, when multiplied by 6, produces ac. Similarly, 4b lg Se age a2 4 FW 87. To divide a fraction by any quan- tity,is to multiply its denominator by to multiply c by If we take a.c, product, therefore, we must take that quantity. Dividing = by ec, the quotient is 7 for, multiplying this quotient by c, the divisor, the product [85] is — or - , [79] the dividend. ss. To multiply one fraction by another; Take the product of thewr numerators for the numerator of their product, and the product of their de- 23 nominators for its. denominator. To multiply ; by 5 is, by [86], to multiply ; by ¢, and divide it by d. The result, fe bd LSet on G =a gk by [85] and [86], will be 3 a) So : x- 6 2 both , a Cc c 89. Since —— x —, and— CO ae b d ac 122, } g equa ba it follows that a Cc Cc Gouge i veshg bE ae We may there- fore extend to fractions the proposition in art. [23], which was proved there for whole numbers only. 90. Any power of a fraction has the numerator raised to that power for a numerator, and the denominator raised to the same power jor a denominator. For instance, (=) 4 = - ‘ = : or - Similarly, (5 )'= - and (= ae 91, Let us next propose to divide any quantity, as a, by a fraction, as Z We ; b : seek an expression for a> a Mullti- ply the divisor and the dividend, both by c, and this becomes acc. re ac A ply any quantity, then, by a fraction, multiply it by the denominator of the fraction, and divide the product by tts or, a on b; or, finally, To multi- numerator. Thus, 9 10 500 5 — — = ——. 50, —, or 55 —. ep MECH SMD 92. When the dividend is a fraction, multiply the numerator of the dividend by the denominator of the divisor Sor the numerator of the quotient, and the denominator of the dividend by the numerator of the divisor for its deno- d a ah y y Re Se —,b minator. Thus, ace wah x y ad ee 24 [oi], sat be 80> 6.) sa tea 24 93. Observe, that ia b a —— means —-, Cc be —— CC = b a by [87]; that a b ac (2) means a rage or? by ror]; that oP Pacer ad Gy Sere: d by [92]5. that * . 2 (one _ ab Pas a * a a ad 94. To multiply a mixed number by a fraction, or by a mixed number, or to divide a fraction or a mixed number by a mixed number, Reduce to improper fractions, and then divide or multiply by the rules laid down. For instance, 3 1 43 7 10, ear Nee eee pel Oe Se 4: 4 9 4 9 36 gt 1 1 erste ba (Bs tad i 5 7 5 7 31 7 217 67 —= =X — ——— = —__ a " Ce 95. When we multiply any quantity by a proper fraction, we have to divide it by a quantity greater than that by which we multiply, and therefore the product is less than the multiplicand, When we multiply by an improper frac- tion the product is greater than the multiplicand. Again, when we divide by a fraction, if it be a proper fraction, the quotient is greater; if it be an im. proper fraction, the quotient is less than the dividend. When the multiplicand is the same, the product increases when the multiplier is increased, and dimi- nishes when it is diminished, When the dividend is the same, the quotient di- minishes when the divisor is increased and increases when it is diminished. 96. Unity divided by any quantity is ealled the reciprocal of that’ quantity. ARITHMETIC AND ALGEBRA. Pee ; om Thus mis the reciprocalofm. 1 se . = = is the reciprocal of = By [91], to divide by any quantity is the same thing as to multiply by its reciprocal. Any quantity multiplied by its recipro- cal, must be 1. 97. The sum of two irreducible frac- tions whose denominators are prime to each other, can never be a whole num- ber. Thus, = + = cannot possibly be a whole number. For, let the fractions be = and = and let their sum be p. Then a al OB: oe ye =p. Multiply each of these equal quantities by 6, and the products will be equal. Therefore a b eae ae bp. a! ma irreducible fraction, and 0 is prime to 0’, by supposition; therefore, by [61] and ab He is an irreducible fraction. It follows that 5b p cannot be a whole number ; for a whole number cannot be equal to the sum of another whole number, and an irreduci- ble fraction: and since p bd is not a whole number, p cannot be a whole number. It is a consequence of this, that if there be any set of irreducible fractions, and the denominator of one of them be prime to all the rest, their sum cannot be a whole number. Of Compound Numbers. 98. Instead of expressing the magni- tude of a thing by means of one unit and its fractional parts, it is usual to have for each kind of thing a scale of units of different magnitudes. We state how many times the unit of the greatest magnitude not greater than the thing in question is contained in it, then how many times the next greatest is con- tained in the part over, and so on to the least. In this way we avoid the inconvenience of having only a small unit, which would often make ‘it ne- cessary to employ yery large numbers, Now a’ is prime to 0’, because is an [62], a’ bis prime to 0’, and : ARITHMETIC AND ALGEBRA. as well as that of having only a great one, which would incumber our opera- tions with fractions. Units of the same kind but of different magnitudes, as pounds, shillings, pence, and farthings, which are units of value, or miles, fur- longs, poles, yards, feet, and inches, which are units of length, are called units of different denominations. A magnitude expressed in units of differ- ent denominations is called a compound number. 99. Numbers are changed from one denomination into another by the rules of reduction. To reduce from a higher denomination into a lower, Multiply the number of the highest denomination by the number of times that the unit of the second denomination is contained in that of the highest, and to the product add the number of the second, tf any ; multiply this result by the number of times that the unit of the third denomi- nation ts contained in that of the second, and add the number of the third, if any ; proceed in this way till the num- ber is reduced to the denomination re- quired. To reduce 5/. 8s, 6d. to pence, for in- stance: since there are 20s. in a pound, 5/. 8s. are equal to (20 x 5+8)s., or 108s.; and since there are 12d. in a shilling, 108s. 6d. are equal to (12 x 108 + 6)d., or 1302d. 100. To reduce a fraction of a unit of a higher denomination into a lower, Multiply the fraction by the number of times that the unit of the lower deno- mination is contained in that of the higher; the product is a fraction of the lower denomination, and if an im- proper fraction, may be reduced into a miaed number of that denomination. For example, to reduce ¢ of a yard into feet and inches. There are three feet in a yard, therefore ¢ of a yard are 18 4 of 3 feet, or ae of one foot, or 24 feet. Again, # of a foot are # of 12 inches, or So that 3 7 of a yard = 2 feet 62 inches. 101. To reduce a number from a lower denomination to a higher, Divide at by the number of times that the unit in which it is expressed is contained in that of the next higher denomination, noting the remainder ; divide the whole part of this quotient again by the num- ber of times that the unit of its present denomination is contained in that of the next higher, noting the remainder, as of one inch, or 6% inches. 23 before; proceed in this way till the number is raised to the denomination required ; the several remainders are the numbers of the several lower deno- minations. For instance, to reduce 87431 seconds to hours. Since 87431 = 60 x 1457 + 11, and there are 60 seconds in a mi- nute, 87431 seconds = 1457 minutes, 11 seconds. So, since 1457=60x24+17, and there are 60 minutes in an hour, 87431 seconds are 24 hours 17 minutes 11 seconds. 102. To reduce whole or fractional numbers of lower denominations into fractional parts of a higher; Reduce the numbers into their lowest denomina- tion, and divide the result, whether it be whole or fractional, by the number of times thai its unit is contained in the unit of the denomination into which the whole is to be reduced. Thus, to reduce 7s. 42d. to a fraction of apound; itis the same as 882d. [art. 99]. But a penny is 34, 0f a pound, and 883 443 therefore 882d; = ——, or’ —— Beet: 24010 200 Bee pound, 103. To add compound numbers to- gether, Write them under one another, the numbers of the same denomination being all in the same column; take the sum of the column of the lowest denomina- tion, and reduce it as far as practicable into the next higher; write under the column of the lowest denomination the part of its sum that ts of that denomi- nation, and carry the other part to be added to the next column; the sum of this column to be treated in the. same way, and so on to the column of the highest denomination. To subtract one compound number from another, Write the number to be subtracted under the other, as in addi- tion. When a number of any denomi- nation in the upper number is less than the one corresponding to tt in the lower, add to it as many as will make one unit of the next higher denomination, and encrease the number of the next higher denomination in the lower number by unity; then proceed to subtract the lower number part by part. 104. To multiply a compound num- ber by a simple number, Multiply every particular part of it by the simple num- ber, and reduce the products as far as as practicable to higher denominations. This operation, however, is much better performed by the rules of practice, ex- plained in the next article. 26 To divide a compound by a simple number, Divide the number of the highest denomination by the divisor, and write the whole part of the quotient as the number of that denomination in the general quotient ; reduce the remain- der, if any, to the next lower denoma- nation, and adding the number of that denomination, if any, proceed as before. These rules are founded on the same principles as those for the addition, sub- traction, multiplication and division of simple numbers in arts. [20], [21], [33] and [40]. 105. Suppose we have to multiply Qs. 6d. by 36, we may multiply its parts separately, as is directed in the last article. But the operation will be much abridged, if we notice that 2s. 6d. are 5 of a pound ; that therefore we have to find 36 times 4 of a pound, or § of 36/., which is 43J., or 4/. 10s. Again, if we have to multiply 3s. 9d. by 17%, we observe that 3s. 9d.= 2s, 6d. + ls. 3d., or 3J. + +41. Therefore 174 x 3s. 9d. = 174 X G + AL, or & + 3) x 1744, or ( + ge) X V7. 10s. Therefore, if we divide 17/. 10s. by 8, and then by 16, or, which is the same thing, divide 17/. 10s. by 8, and the quotient by 2, and then add the two quotients, their sum will be the answer. We haye ‘ 1. SS8rbwe 8)17 10 0 {Sie aoa On the same principles, fo multiply 1/. 14s. 3d. by 173, is to take the product “GQ+4+4754¢+ H+ »D xX 1731, as appears below : l. s. a ye Sa, LOO oY ollie Ord POLL = 049 34 12 0 0 40 34 12 0 0. 4°90 2) 34°12 0 A of4s.=0 20 8] 17 60 Lof2s.=0 03 2°33 1143 296 53 Similarly, to multiply 2 feet 10 in. by 113, 1s to take the product Git atistds) x11 yds. 2 ft. 3 in, Since 2 yd. = 2 feet 3 in. P yds. ft. in. vidi ft: dn. 8. ins 112° 8 Dyn OE Ieee BES i190: LSD bay ARITHMETIC AND ALGEBRA. | The submultiples [50] of a unit of a higher denomination, when they can be expressed in whole numbers of lower de- nominations, are called the alzquot parts of that unit. Thus, 10s., 6s. 8d., 5s., 4s., and 3s. 4d., which are respectively, dete Blast 1 —, —, —, — and — of a pound, are Bt 8 ea oe 6 aliquot parts of a pound. So 1ft. 6 in., ° 1ft. 9in., 6in., &c., are the aliquot parts of a yard. Every compound number of lower denominations can be reduced in many different ways into aliquot parts of a unit of a higher denomination. A little familiarity with the rule of prac- tice, by which we have resolved the foregoing questions, teaches us the most convenient set of aliquot parts into which to reduce anynumber. The rule is this: Reduce the compound number to be multiplied into a sertes of aliquot parts of a unit of some denomination, such, that every aliquot part is either the same as the preceding, or some sub- multiple of it; consider the multiplier as a number of that denomination, with respect to which the aliquot parts are taken, and reduce its fractional part, of any, into lower denominations ; divide the multiplier, in this state, by that number by which the first aliquot part is a submultiple of the standard unit ; divide the quotient by that number by which the second aliquot part ts a sub- multiple of the first; proceed in this way till all the aliquot parts are ex- hausted, observing to repeat a quotient twice or more, when two or more aliquot parts arethe same; add all these quotients, and their sum ts the product sought. 106. The process of multiplication is the addition of a quantity to itself a cer- tain number of times, so that whatever the multiplicand be, the multiplier must always be an abstract number, and the product a quantity of the same nature as the multiplicand. It is therefore an absurdity to speak of multiplying such quantities together as pounds and yards, or pounds and pounds, or yards and yards. When there are so many yards of cloth, for instance, at so much a yard, and the price of the whole is re- quired, to find it, we do not multiply money by yards, we take a certain sum of money as often as there are yards of: cloth, the sum, of course, is a sum of money. Again, if a pound will buy a certain number of yards, and we have a certain number of pounds, and wish -to know how many yards we can buy, we take the number of yards as often as there are pounds, and the sum is a ARITHMETIC AND ALGEBRA, number of yards. There is one seeming exception to this, namely, that we fa- miliarly multiply feet by feet, or inches by inches, and have for product square feet or square inches. But this only needs a little explanation. Suppose, for instance, that there is a square board, 8 inches each way, to find the size of its surface, we are said to mul- tiply 8 inches by 8 inches, and to have for product 64 square inches. But the proper way to consider the matter is this; suppose the board to be divided into squares, so as to resemble a draught or chess board; then, because there are eight inches along the end, there are eight squares in every row; and because there are eight inches along the side, there are eight rows of squares. All that we do, then, in multiplying 8 by 8, is taking 8 square inches 8 times, and adding them together; and, strictly speaking, this is not multiplying inches by inches, any more than if there were a shilling on each square, it would be multiplying shillings by shillings, to find their num- ber by multiplying the number in every row by the number of rows. We cannot, therefore, properly speak- ing, multiply one compound number by another.. When we have such a ques- tion as to find the value of 8 lbs. 9 oz., at 7s. 6d. per lb., we must consider, that, as there are 16 oz. in a pound, 9 oz. are 2; of a lb., and therefore cost 5°, of the price of a pound; and so, to find the answer, multiply 7s. 6d. by 875. In such cases, a little attention will always teach us which of the compound num- bers is to be reduced to a simple num- _ber before we multiply. The rule of duodecimal multiplication, which is another way of multiplying compound numbers of certain kinds, depends on principles that cannot properly be ex- plained here. 107. The product of a compound number by a simple number, is, as we have seen [106], a compound number of the same nature as the multiplicand, and we have also seen [10], that the division of the product by the multiplicand gives us the multiplier for quotient. We may therefore properly divide one compound number by another of the same nature, for in so doing we seek the simple num- ber which must multiply the divisor so as to produce the dividend. The most convenient way of performing this di- vision, is to reduce the divisor to its least denomination, and proceed as is directed for a simple number [104]. 27 Of Simple Equations. 108. The sum of two numbers is 89, and their difference is 31; let it be re- quired to find out what the numbers are. Call the less number » Then, since their difference is 31, the greater is 31+. Also, their sum is 89, there- fore w [theless] +(31+ 4) [the greater]=89, that is, adding the greater to the less Zid! A'S WS B9 If from each of these equal quantities we take away 31, the remainder will be equal. Therefore 2H = 89 — 31, or 2% = 58, And dividing each of the equal quanti- ties by 2, w = 29, and therefore 31 + # = 60. We find the one number to be 29, and the other 60; and these two numbers satisfy the conditions that were given. In the same way, if the question had been to find two numbers whose sum is a, and difference 6, calling the less #, we should have found. 24+b=a, id pent The numbers are ~~ Pies EB : & In these expressions any numbers may be substituted for a and 6, and so an answer found in any particular case. _ Two expressions connected by the sign of equality, as in these examples, make an equation. The two expres- sions so connected are called the two members of the equation. ~An equation in which the quantity whose value is sought, occurs in the first power only, is called a sample equation ; a distinction of which the importance will appear afterwards, 109. In the example in the last article, we removed the number 31 from the left hand member of the equation to the right, changing its sign. In the same way, in any equation, if a term be struck out of one member, and written in the other with its sign changed, the new expressions so formed will be equal to each other. If in this way we remove every term from the left hand member into the right, and every term from the right hand member into the left, the two members will have the same terms as before, only the sign of every term will 28 be changed. Thus, if a—-nx=mzr—b, we should have | b—-mr=nxv—a. We may therefore change the sign of every term in an equation, and the new members will still be equal to each other. If we remove all the terms from one member into the other, we shall have zero on one side of the equation, Thus a—-nx=mxz—b becomes atb—(n+m)r=0. The meaning of this is, that a + 6, and (n + m) x, are two quantities such that their difference is nothing, that is, they are equal quantities. We shall after- wards find, that this way of stating an equation is sometimes very convenient. 110. Again, let it be required to find that number, the third part of which added to its seventh part makes 20. Let the number be called x, as_ before. Its third part is - and its seventh is - Therefore i x ee ie 203 3 as 7 Multiply both members by 21; then, since the products must be equal, Vet 3.074205 and adding, as before, 10 2 = 420. Now divide each member by 10, and we find x =42. The third part of 42 is 14, its seventh part is 6, and these added make 20. When there are fractional terms in an equation, we can always get rid of them, as in this example, by multiplying both members by the product of all the denominators, or by their least common multiple, if it be less than their pro- duct. By [80] all the fractions may be reduced to a common denominator, which will be this common multiple, and multiplying all the terms by it, the fractions will disappear. 111. These examples teach us how a simple equation is to be solved, that is, how we are to extract from it the value of the unknown quantity. Clear both members of fractions, rf there be any, [110]; collect into one member all the terms containing the unknown quan- tity, and into the other all those that do ARITHMETIC AND ALGEBRA. not contain it; collect into one the co- efficients of the terms containing the unknown quantity, divide both mem- bers by the whole coefficient of the un- known quantity, which will then stand alone in one member. We may observe here, that a simple equation can have only one solution. By the rule, every simple equation can be reduced to the form, Axvzt+B=0. If a be a value of a that solves it, we have Aa + B= 0. Suppose, if possible, that a’ is another value of x that solves it, then Ad+B=0. Subtract A a’ +B from A a+ B, the difference must be nothing, so that A.(a—a)=0. ‘But a product can be nothing, only when one of its factors is nothing. So that a— a = 0, that is to say, a’ cannot be a solution, but on the condition that it is equal to a. 112. We will now apply this very simple rule to a few questions, with some remarks on the results which we shall obtain. Take the question pro- posed in art. [3]. Let the time which the father and son take to dig the field together be called a The father in one : il day digs Li of the field, in two days ; : 2 ‘eS he will have digged o of it, in x days he will have digged — of it. So the son in @ days will have digged = of the field. But in x days they will have digged the whole field. Therefore Xx } x eh of the field + oe of the field = the whole field. The magnitude of the field is a quantity which is a factor in every term of this equation, dividing each member by this quantity x wo To a re 1. Whence, by the rule, multiplying by 80, the least common multiple of 10, and 16,4 2 a? whence w = 8. ARITHMETIC AND ALGEBRA. 29 : Su+5x# = 80, and Similarly, if the father’s time were a, and the son's 4, we should find agp tld ; a betiee2? and ab ato With regard to this expression for #, observe that a enters into it in the same way as 6 does; a therefore might be written for }, and 6 for a, without chang- ing its value. Such an expression 1s said to be symmetrical with respect to aand 6. This must necessarily be the case fromthe question ; for if the father took 5 days, and the son a, to dig the field, the answer would be the same. An examination whether an algebraical result be symmetrical with respect to quantities that enter into the conditions of the question in the same way, often enables us to detect errors. 113. A father is 40 years old, his son is 8; in how many years hence wiil the father’s age be just three times the son’s? Let the number of years to that time be called 2 In # years the father will be 40 + # years old, and the son 8+.w. But the father’s age will be then three times the son’s; therefore 40+#v=3(8ta=24 +3. Carry « to the one side of the equation, and 24 to the other, and it will become 40° — 24 = 3 —2@, or 16=22; The father’s age at the end of 8 years will be 48, and the son's If the father’s age had been called a, and the son’s 6, to answer the same question we should have had, as before, atw=3b+ 3a, 2%=a—36, a—3b- 2 ? an expression which gives a value of w for every different value of a and 6. Suppose that the father’s age is 40, and the son's 18. Here the time when the father’s age was three times fhat of the son's is already past, so that the ques- tion, if put as it stands above, would be absurd. Let us see what our expression for # becomes in this case: making a= 40 and b = 138, 40—3x18 40—54 _ _ — 2 2 a negative quantity. In our original conditions, the father’s age was to be 40 +4, and the son’s 18+ 2. These are in the present instance 40 — 7, or 33, and 18 ~ 7, or, II, | And. 33 is'2 times 11. The negative sign teaches us that the number of years that 1s to make the father’s age 3 times the son’s is to be subtracted, not to be added, and ac- cordingly we find, that 7 years ago the father’s age was just three times the son's. Observe, that when a= 36, x=0; that is to say, that when the father is three times as old as the son at the pre- sent time, z is neither a quantity to be added nor subtracted. This question may be made still more general by proposing to find when the father’s age will be 2 times the son’s. We should have as before atwv=nb+nz, whence we find a—nb m—1° 114, A and B find a purse with shil- lings in it, A takes out two shillings and one sixth of what remains ; then B takes out three shillings and one sixth of what remains ; and then they find that they have taken equal shares. How many shillings were in the purse, and how many did each take? Let x be the number of shillings in the purse at first. A takes out 2s.; there remain x — 2. 9 by em 1 : : _ He takes z of this remainder, or “a ? eh : there remain a7? of the first remainder, or 5x2— 10 6 e B takes out 3s.; there remain 5x2— 10 5x2 —28. ——_———- — 3, or ————_- 6 6 1 : ‘ He takes —— of this remainder, or xv —2 6 5 xv — 28 B has taken out 3+ Peoe oR therefore since their shares are equal Now A has taken out 2 + , and and 30° 52, — 28 S6 i Multiply both members by 36, and they become 72+6—12=108+ 52% — 28. And collecting the terms w= 20 c—2 Mamas co Pysr There were 20s. in the purse. A’s tx—2 20. — 2 share was 2 + ,or2 + a 5a = 28 or 5s.; and B’s was 3 + ———, 36 which will be found to be 5s. also. 115. There is acertain number consist- ing of two digits, and theirsumis 6. If 18 be added to the number the sum will consist of the same digits, im an in- verted order. What is the number? Let its tens’ digit be called x, then since the sum of its digits is 6, its units’ digit will be 6—a. The number then is 10“ +6—a4. Nowif 18 be added to the number, its units’ digit becomes 2, and its tens’ 6 — a, and therefore it be- comes 10(6—a#) +a. Therefore 104 + 6 —# [the original number] +18 = 10(6 —#) + w[the new number]. And collecting the terms 1520 =2 ODS UC 2 6 The digits are 2and 4, The first num- ber is 24; and 24+ 18 = 42. 116. A hare is 80 of her own leaps before a greyhound; she takes three leaps for every two that he takes, but he covers as much ground in one leap as she does in two. How many leaps will the hare have taken before she is eaught? Call the number of leaps a. Since the dog takes two leaps only for every three that the hare takes, he will have taken = x leaps while she takesa. But since one of his leaps is equal to two of hers, in these = x leaps of his he will have covered as much ground Now 2 z @, OY rs w hare’s leaps as 2. #,0r-. ps. when he has done this, he catches the . hare, by our supposition, and he was 80 of her leaps behind her at first, there- fore he has run a distance equal to 80 +a of her leaps. We have thus found two expressions for the distance ARITHMETIC AND ALGEBRA, run by the dog before he catches the hare, which of course must be equal. So that we have i v=80+a 3 ie 3 and multiplying by 3, 4-&@ = 240 + 32, whence & = QA, Suppose that, all other circumstances being the same, it had been said gene- rally, that m of the greyhound’s leaps were equal to 2 of the hare’s. Then 22 his ME leaps would have covered as Be much ground as ~ : ei of the hare’s and we should have had 2n om _ 2N# = 240M + 3mz, (272 —3m)# = 240m, and 240 m ~ on —3m' Now suppose, that m were 3, and 2 3 X 240 8—9’ — 720. Let us inquire into the mean- ing of this‘negative sign. As our ques- tion now stands, the hare is 80 leaps before the dog, she takes three leaps for his two, and three of his are equal to four of hers. In this case it is clear, that the hare runs faster than the dog, and that she is continually getting away from him. In turning our question into an equation, the way in which we were 4, # would become expressed that in x leaps the hare would be caught, was by expressing that at the end of x leaps the hare and dog would be together, which when the dog runs faster than the hare means the same thing. But just as when the dog runs faster than the hare, we look forward for the time when they will be together; so when the hare runs faster than the dog, we must look back for the time when they were together. Ac- cordingly we shall find, that if the chase have lasted for 720 leaps up to the pre- | sent time, it began by the dog and- hare being together, and in 720 leaps’ the hare has gained 80 on the dog. ' This is what is meant by the negative | 145) ARITHMETIC AND ALGEBRA. sign. It is impossible that our alge- braical symbols should contemplate any beginning or ending to the chase. What is called the law of continuity requires that we should consider it as prolonged indefinitely both ways. Once more, suppose that in our ex- pression m were 2,and 7, 3. In that case, we have the dog making two leaps for the hare’s three, while two of his leaps are equal to three of hers. They run with the same speed, then; and if the course be lengthened ever so far either way, they have been, and always will be, at the same distance from each other; so that they never have been, and never will be, together. Let us see how this result is shown by our ex- pression for x. It becomes 240 240 240 a= ———_,, OF ——_.,, OF C Ox 73 x2 66’ 0 When the denominator of a fraction becomes less, the fraction itself be- comes greater. When the denomina- tor becomes very small, the fraction becomes very great; and no quantity can be named so great but that we can make the fraction greater than it, by tak- ing a quantity small enough for its de- nominator. It follows, that a fraction whose denominator is nothing, is greater than any quantity that can be named. The value of 2 just found, is there- fore greater than any quantity that can be named; and this shows that the number of leaps taken before the dog and hare come together is greater than any number that can be named, that is, that they never come together. 117. A quantity greater than any quantity that can be named, as ex- plained in the last article, is said to be uyinitely great. A number infinitely great is called infinity. The alge- braical symbol for infinity is . 118. We see that the main difficulty of answering such questions as these, lies in finding equations to express them, as soon as that is done the solution is easy. The art of turning such ques- tions as occur into algebraical lan- guage, is one for which no general rule can be given; it consists in separating from the question all the circumstances that are not essential to its solution, and can be acquired only by practice and careful thought. Questions that can be solved bya simple equation, and one unknown quantity, can always be answered by one of the- arithmetical rules of single or double position, Of 3} these an account will be given, after we have treated of proportions.* 119. Suppose that there is such an equation as containing two unknown quantities, # and y. For.every different value given to y,# has a different value; so that we can find as many pairs of values of x and y as we please, which, when sub- stituted for them in the left hand mem- ber, make it equal to 43, and which are therefore said to satisfy the equation. Suppose, again, that there is another equation of the same sort, 124% —8y=4; we can, in the same way, find as many pairs of values of # and y to satisfy it as we please. Now, of all the pairs of values that satisfy the one of these equations, there is one, and but one, that will satisfy the other. It may be found in this way. Multiply both mem- bers of the first equation by 8, the co- efficient of y in the second, and both numbers of the second by 7, the co- efficient of y in the first. They then become 40¥+ 56y = 344, 84v—56y= 28. If equal quantities be added to equal, the sums must be equal; the sum of the left hand members of these equations is therefore equal to that of the right; and as we are now supposing w and y to have the same value in the one equation as they have in the other, these sums are 124 @ = 372; whence # = 3. Substituting this value for # in the first equation, it becomes 15+ 7y = 43, which gives EG? It is impossible that any pair of values of w and y, other than 3 and 4, can satisfy both equations: for no values can satisfy both equations that do not also satisfy the equations 124@=372,and15+7y=43, and we know that there is but one value that can satisfy a simple equation, contaiming only one unknown quantity. CLE Ee 120. This furnishes us with a general rule for finding values for two unknown * See art. 138, 82 quantities that will satisfy two equations containing them. Clear the equations of fractions, if there be any, and in each of the equations collect the co- efficients of each of the unknown quan- tities into one; fix on one of the un- _ known quantities, and multiply all the terms of each equation by the coefficient of that quantity in the other; that unknown quantity will now have the same coefficient in both equations, and by the addition or subtraction of thetr members, according as this coefficient has different or the same signs in the two equations, they will be reduced to one equation containing one unknown quantity ; when the value of this un- known quantity is found, substitute it in one of the original equations, which will then contain the other unknown quantity only. 121. I have a certain number of counters in each hand; if I put ten out of my left into my right, there will be twice as many in my right as remain in my left; if I put ten out of my right into my left, there will be three times as many in my left as remain in my right ; how many are there in each hand? Call the number in the right z, and that in the left y. When I put ten out of my left into my right, these numbers become # + 10, and y — 10; and when I put ten out of my right into my left, they become a — 10, and y + 10. Now by the question v +10 2(y — 10), and y +10 = 3(@ — 10); or, multiplying and collecting the terms, 2y — x= 30, 3a—y = 40. Multiply the first of these equations by 3, it becomes 6y —3H4# = 90, To this add the second, and we find 5 y = 130, whence y = 26. Substitute this value for y in the second equation, it becomes 34 — 26 = 40, whence w = 22. So that 22 in the right hand, and 26 in the left, are the num- bers which satisfy the conditions given. 122, Just in the same way, when there are three unknown quantities, and three equations containing them, we can find one set of values that will satisfy every one of the three equations. This 1s done on the same principles as when there are two unknown quantities. ARITHMETIC AND ALGEBRA. A, B, and C, sit down to play, every one with a certain number of shillings. A loses to B and C as many shillings as each of them has. Next, B loses to A and C as many as each of them now has. Lastly, C loses to A and B as many as each of them now has. After all, every one of them has 16 shillings. How much did every one gain or lose? Call A’s first sum 2, B’s y, and C’s-2. First; A loses y to B, and z to C. He has remaining «— y — 2. B has 2 y, and C has 2 z. Secondly; B loses to Aw —y —2, and to C 22. He loses altogether e—y—Z+22,0rv—yt 2. Anowhas2#—2y—22, Bhas2y —(@ —y + 8), 3Y — kX — @. © has 4 2. Lastly ; C loses to A 2#—2 y—2 2, and to B3y—a#—z. He loses in all 2u—2y—22+ (3y — # — 2), or Wire oe A now has4a—4y— 42. Bhas6y—24—22. Chas4z—-(e#+y —32), or 72—-XH—Yy. Now, at last they all have 16. So that A’s sum, or4 vx —4y—42= 16, whence e@—y—2z=4,...(@ B’s sum, or6¥ — 2% —22= 16, whence 38y—# —z=8,..(0) C’s sum, or 72—H# —y=16..0 Add (a) and (8), then 2Qy—2z2= 12...) Add (a) and (c), then 6z2—2y = 20....© Add (d) and (e), then 4z= 32, whence z = 8, Substitute 8 for 2 in (), it becomes 2y — 16 = 12, whence y = 14. Substitute 14 and §& for y and z in (a), it becomes w—14—8 = 4, whence #« = 26. So that A’s origina! sum was 26, and he has lost 10. B's original sum was 14, and he has won 2. C’s original sum was 8, and he has won 8, Po or ARITHMETIC AND ALGEBRA, 123. We have seen [119], that we may find as many pairs of values of two unknown quantities as we please, that will satisfy one equation that con- tains them both. In the same way it may be shown, that we can find as many sets of values as we please of three unknown quantities that will sa- tisfy one equation, or each of two equa- tions containing them all; and so for a greater number of unknown quanti- ties. It follows, that when we make use of two or more unknown quantities, to solve any question that admits of one value only for each unknown quantity, we must always obtain as many equa- tions as there are unknown quantities. Again, in solving such a question, if we obtained one equation between two unknown quantities, and formed another by multiplying all its terms by some quantity; or if we obtained two equations between three unknown quantities, and formed another by add- ing these two together, the new equa- tion so formed would be of no service. It could give us no new information with respect to which of all the pairs of values of the two unknown quantities that satisfy the single equation, or of all the sets of values of the three un- known quantities that satisfy the two equations, is the pair of values, or the set of values required by our question. The new equation is not an independent one. The reader will find on trial, the impossibility of reducing two equations between two unknown quantities, to ‘one equation containing one unknown quantity, when these equations are not independent. We shall return to this when wetreat of¢ndeterminate problems. When there are more equations than there are unknown quantities, it may be impossible to find one set of values that will satisfy them all. Suppose, for instance, three equations between two unknown quantities. We can find, as we have seen [art. 120], a pair of _ values that will satisfy the first and second, and also a pair that will satisfy ‘the first and third; but it is a mere chance if the two pairs of values so found be the same. Of Proportions. 124. Let there be two sets of num- bers, such as : 1 9,21, 33, 40, 60, 94-, 297.... 2 6, 14, 22, 2635 4030 G3; 4.198). 5% 33 The numbers in the upper line are any whatever; those in the lower are so taken, that when a number in the upper line is divided by the one under it in the lower, the quotient shall al- ways be the same, as in the present in- 8 - stance 3 When this is the case, the numbers in the lower line are said to be tn direct proportion, or directly pro- portional, to the corresponding ones in the upper. Since the quotient, when a number in the lower line is divided by the one above it, is the reciprocal [96 ] of the quotient when the number in the upper line is divided by the one below it, that quotient also is always the : ; s,HaraD same; in the present instance it is a Therefore the numbers in the upper line are also in direct proportion to the corresponding ones in the lower. 125. Daily experience furnishes us with sets of numbers that are in direct proportion to each other. Thus, if the lower line contains different numbers of yards of cloth of the same sort, and the upper their respective prices, the numbers will be in direct proportion ; for the price divided by the number of yards, gives the price of one yard, which is the same, whatever be the number of yards. So if the one line were numbers of miles travelled at the same rate, and the other the respective numbers of hours spent in travelling, the numbers in the respective lines would be in direct proportion ; for the number of miles divided by the corresponding number of hours, gives the number of miles travelled in one hour, or the rate of travelling, which is uniform. To find out whether any two corre- sponding sets of numbers are in direct proportion, we must. consider whether we are sure that the quotient of two corresponding numbers is the same for all. If we do not see a reason, as in the cases just referred to, why it cannot be otherwise, we must not con- clude that they are in direct proportion. If one set of numbers, for instance, ex- pressed the different sizes of barrels, every one with a hole in it of the same bore, we have no right to suppose that another set of numbers expressing the minutes that every barrel takes to run out, will be in direct proportion to the first; for we see no reason why all the quotients must be the same. In cases like this, where we do not see our way D 34 clearly, other parts of the mathematics can be employed to find out whether the numbers are or are not in direct proportion ; and if not, to find in what way they do depend on each other. 126. If we examine attentively what we mean when we speak of the propor- tion which one thing bears to another of the same kind, we shall find that we mean the magnitude of the quotient, when the number expressing the mag- nitude of the first thing is divided by that expressing the magnitude of the second. We call the proportion great when this quotient is great, and small whenit is small. Thus, 12 bears to 4a greater proportion than 18 does to 9; be- T2¢ 18 cause — is greater than co So 2 bears to 11 a less proportion than 7 does to hell 7 23; because — is less than —: and 15 11 23 bears to 5 the same proportion as 24 : 15. 24 does to 8; because =e equal thers The proportion that one quantity bears to another, is often called its ratio to that other. From our defini- tion of direct proportion, it will be seen that when there are two sets of quanti- ties, of which every pair of corre- sponding quantities have the same pro- portion or ratio to each other, the quantities in the two sets are in direct proportion to each other. 127. Suppose, now, that a and 6 are corresponding quantities in two sets that are in direct proportion, and that a and 0b’ are other two; then by art. [124] a a! = rae This relation between these quantities is often written in this way, Boo 4 Og and this is read, a isto 6, asa’ is to b'; that is, a has to 6 the same ratio or proportion as a’ has to 6’. From their situations in this way of stating a proportion, aand 6’ are called extremes ; 6 and a’ means, thatis, middle ones. 128. If we multiply each member of the equation in the last article by 6 U, it becomes av =a! b. The product of the extremes then is al- ways equal to the product of the means. Thus, since / Tig 2 we have 12:9, 4x9=3-x Iz, ARITHMETIC AND ALGEBRA. Again, if we multiply each member of the same equation by GW it becomes a b a’ Be whence Beal fe Phy shel Ak showing that the first has the same ratio to the third, as the second has to the fourth. Thus, since EPG FW Bee ® we haye - Opt Cee al Olas Again, if unity be added to each member of the same equation, it be- comes a a Rice sik: baa or a+b ad+0 b Bri That is, at b:bird + U8. And similarly by subtracting unity, ab *t a =e So that the sum or difference of the first and second has to the second the same ratio, as the sum or difference of the third and fourth has to the fourth. Thus, since FINS oe Los and since 7 + 3 = l0and 35 + 15 = 50, we have ; 10352" 501550 Also, since7 —3 = 4and 35 — 15 = 20, we have ; BE Tiiee'4 Ue Bic Quantities in direct proportion have many other properties, all of which can be easily deduced from the equation a a’ ara 129. When we have a ae, O Cc b 3 or Gd: 2p, where the two means are the same, 3 is said to be a mean proportional between a and c, and ¢ is said to be a third proportional to a and ba: is also said to have to c the duplicate ratio of ato 6. The last term is not a very well chosen one; the ratio of ato ce is considered as if it were made up of the ratios of @ to band of 6toc; now the ratio of Bb to c is the same as that of a to 3, there- fore the ratio of a toc is the ratio of a to 6 two-fold, or the duplicate ratio of ato 5. In this case a, 6, and c are sometimes ARITHMETIC AND ALGEBRA. said to bein continued proportion. When three quantities are in continued propor- tion, the product of the first and third is equal to the square of the second. This follows from the first property proved in art. [128], which becomes in this case ac = 6, Similarly, the quantities a, 6, c, d, e, &c. are said to be in continued propor- tion, when fe Cc d and just as before, a is said to have to _ d the triplicate ratio that a has to b; ais said to have to e the quadruplicate ratio (or fourfold ratio) that it has to b, and so on, Since Oe. a eu ieige multiplying both members of this equa- : a _ tion by Fz we have ae Cun oe Again, multiplying the first member of the last equation by > and the second by a Oe F which are equal quantities, we have a aa igh. ADR? and in like manner a at pet and so on. 130. Let us next have two lines of numbers, such as +t Beis ht, ibs, 2 6>- a0, 1] 2@e00 1 1 180, 90, 60, wire dad, Ae osea where the product of any two corre- sponding numbers is always the same, in this instance 180. When this is the case, the numbers in the one line are said to be inversely or recipro- cally proportional to those in the other. The numbers in the one line are in fact directly proportional to the reciprocals [96] of those in the other. For let a and 6 be two correspond- Ing numbers. The reciprocal of a is i vay) 5 hd a and this divided by 6 is ae: Now &@ bis, by supposition, the same as the 35 product of any other pair of numbers. Therefore on dividing the reciprocal of any number by the corresponding num- ber, the quotient is always the same ; that is, by [124], every number is di- rectly proportional to the reciprocal of the correspending one. When two sets of numbers are in direct proportion, those in the one set increase when those in the other do, and at the same rate; if a number in one set be doubled, the one in the other set is also doubled. When they are in inverse proportion, those in the one set diminish as those in the other increase, and that too at the same rate ; if a num- ber in the one set be doubled, the one in the other is halved. 131. As in the case of direct propor- tion, we must never conclude that sets of corresponding numbers are in inverse proportion, unless we can prove that the product of every pair of them must be the same. If one set contains the different numbers of labourers that may be set about the same piece of work, and another set the numbers of days that the respective bands of labour- ers would take to finish it ; the numbers in the one set are inversely proportional to those in the other: for the number of days multiplied by the corresponding number of labourers, gives for product the number of days’ labour of one man required to do the work, which will be the same whatever be the number of labourers. So if one set of numbers contains the hours taken by different persons to travel the same distance, and another the number of miles that every person travels in an hour, the corre- sponding numbers will be in inverse proportion; for the number of hours multiplied by the space travelled in an hour, gives the whole distance travelled, the same for all. 132. If a and a’ be two numbers, and 6 and 0! two corresponding ones in in- verse proportion, we have ab=a Ub. From this we find a a’ Pitt or Oe They are in direct proportion, then, if 6 and 0! change places. On the other hand, if there be two numbers, and other two in direct proportion to them, if the second two change places, they will now be in inverse proportion to the D2 36 first two; and this is the reason why the sort of proportion of which we are now speaking is called inverse. 133. When two numbers are given, and a third, it is the business of the rule of three im arithmetic to find a fourth, such that the second and fourth shall be either in direct proportion to the first and third, or in inverse, as the question may require. The first and third must both represent things of the same sort, as, for instance, sums of mo- ney or labourers ; the second and fourth must also be of the same sort, as yards of cloth or hours. As in the former cases, let a and a’ be the first and third, 6 and 0! the second and fourth. When the proportion is direct, we have [art. 127] Oo Se O's and the equation in the same article gives y 22 a which shows that The fourth is the product of the second and third divided by the first. When the proportion is inverse, we have [art. 132] a: pea =, and the equation in the same article gives ab al which shows that The fourth is the product of the first and second divided by the third. Whether the question furnish us with numbers that are in direct proportion or inverse, can, as we have seen, always be found out by a little consideration. For example, if a garrison of 800 men victualled for 90 days be reinforced by 300 men, for how many days is it now victualled? Here 800 menand 1100 men are the first and third quantities; 90 days and the number of days required, the second and fourth. The numbers of days are inversely proportional to the numbers of men ; for the number of days must be such that the product of it, by the number of men, shall be the number of daily rations of food in the garrison for one man; and the number of these rations is the same after the reinforcement as it was before. There- fore by the second rule, we have 800 xX 90 5 =05°——e 1100 rr ! — ’ Days required = ARITHMETIC AND ALGEBRA. _ 134. Questions in compound propor- tion are those in which five quantities are given, and a sixth is required, or in which seven quantities are given, and an eighth required ; and the like. Such questions can always be answered by reducing them to one of the two kinds of simple proportion. For instance ; if 20 men weave 84 yards in 6 days, how many days will 12 men take to weave 100 yards? Call the number of days sought a. Since the number of yards woven in a given time is directly pro- portional to the number of weavers, and since 20 men weave 84 yards in six days, 12 x 20 or 240 men will weave 12 x 84 or 1008 yards in six days. Also, since 12 men weave 100 yards in x days, 20 x 12 or 240 men will weave 20 x 100 or 2000 yards in x days. So that our question, which stood at first thus, Men. Days. Yards 20 6 84 12 x 100, will stand thus, 240 6 1008 240 x 20005 where the number of weavers is the same in both. This number, then, is now of no consequence to the question, and it may be put: Ifa certaim number of men weave 1008 yards in 6 days, how many days will they take to weave 2000 yards? By the rule of direct pro- portion [133] the answer is 6 x 2000 912 "or 1] —— days. etha TCT ete ys. Observe that aa 20 x 100 x 6 Se AM DR es” 135. In every question of compound proportion there is one quantity given of the same nature as the quantity sought, 2. Call this given quantity, for shortness, a. Of the other quantities given one half belongs to a@ and the other half to zw. The rule of compount proportion is this: Write a, and th quantities belonging to it, in one line, ant x, and the quantities belonging to it below, in another, observing to haw quantities of the same nature one unde the other; when two corresponding quantities have to each other the direc proportion of ato x change their places writing the lower one in the upper line and the upper one in the lower ; divia the product of all the quantities in th upper line as it now stands, including .ARITHMETIC AND ALGEBRA. by the product of all the given quanitr- ties in the lower line, the quotient is x. Thus, in the question in art. [134 ], be- cause the number of yards woven is in direct proportion to the number of days taken to weave them, we make 84 and 100 change places; and because the time required to do a piece of work is in inverse proportion to the number of men employed about it, we let 20 and 12 stand as they are. So that we find x, as before, to be 20 x 100 x 6, 12 x 84 Take another example: If 10 mendig 8 acres in 6 days, working 8 hours a day, how many men will be able to dig 7 acres in 3 days, working 10 hours a day? Writing the quantities as the rule directs, they stand Men. Acres. Days. Hours. (a). 10 8 6 8 x 7 3 10 The days and hours are in inverse pro- portion to the number of men, so that the numbers expressing these quanti- ties must stand as they are. The acres are in direct proportion to the number of men, and therefore the numbers expressing acres must change places. Then, by the rule, by Ge 2. 2G G4 08 SG ide Xe LO Before we multiply we can strike the common factors out of the dividend and divisor. We then find 20 i = 14 men. The truth of the rule in any particu- lar instance can easily be ascertained in the same way as in art. [134]. 136. When the numbers in one set are directly proportional to those in an- _ other, if y stand for any number in the one set and a for the corresponding one in the other, and if m be the quo- tient always proceeding from the divi- sion of a number in the first set by the corresponding one in the other, then Ce) For in that case we shall always have y.. sma — = —- =n. Aaya This relation between y and 2 is some- times expressed by saying that y vartes directly as x. Thus, because the work done in a given time is proportional to the number of workmen, we say that 37 the work varies directly as the number of workmen. So, when the sets contain numbers in inverse proportion, if m be the product af two corresponding numbers we shall ave Y= For in that case RIS ee m eo) ye y xv Here we say, that y varies inversely as a; for instance, the time of doing a piece of work varies inversely as the number of men employed. 137. Besides being directly or in- versely proportional, there are a great many other ways in which the numbers in one set may depend on those in an- other. For instance, take the sets 5, 20, 125, 245, 845,...-(y) Tey Bea A Esso Ge) where the numbers in the upper set are directly proportional to the squares of those in the lower, and consequently increase when those in the lower set in- crease, but at a much faster rate. In this case, any number in the upper line, divided by the square of the one below it, will be found to be 5, so that y= 52% In general, when y = m2*, where m always remains the same whatever values are given to w and y, y is said to vary directly as the square of az. Similarly, ify = = y is said to vary inversely as the square of x. Here y diminishes while x increases, and ata much faster rate. The symbolscc and =, but more usu- ally the former, are sometimes used to denote variation. Thus, that y varies as the cube of x is sometimes expressed y oc a3, But it will always be found much more convenient to use the sign of equality, as we have done above. If this be attended to, such questions as the following can give no trouble. If y varies inversely as x, and u varies as the square of y, in what way does wu vary with respect tov? We have y = =" Also,u =m! y?. In this expression for wu substitute the value m \2 of y, and it becomes wu = m’ (2) or m'me Now mm? is of the nature 38 of m or m/, in respect that it does not vary, and therefore w varies inversely as the square of a. Again, if y varies as the square of 2, and zw varies as x inversely, in what way does the product of y and w vary with respect to x? We have y = m 2 m! m! and wu = —; therefore yu = m2x*. — or uy SO ie mmx. So that yu varies as x directly. 138. When a question produces a simple equation of the form ax= 6b, that is, one in which w does not occur in the one member and is a factor in every term of the other, such a question can be answered by the arithmetical rule of stngle position. Substitute some number, such as s, for x, and suppose that we find that as=JU, Now two fractions are equal when the numerator and denominator of the one are respectively equal to those ef the other ; therefore ax 6b as Oo whence by [79] B b aug or by [127] Ce ae OE Oe The rule of single position is con- tained in this proportion, and is as fol- lows: Suppose some value for the un- known quantity, and find what the result would be on that supposition. The true value of the unknown quantity as to its supposed value as the true result is to the result found. For example: To find the number, of which the half, the third part, and the fourth part added together make 65. .Suppose the number to be 48. The half, the third part, and the fourth part of 48, or 24, 16, and 12 added to- gether make 52. Then we have by the rule 248 2:65 252. And from this we find, by [133], that the number sought is 60. 139. When a question furnishes such an equation as ax+b=ca+d, it can be solved by the arithmetical rule ARITHMETIC AND ALGEBRA. of double position. As in art. [109], this equation may be put in the form (a—c)%+b—d=o. Now suppose that sis substituted for L, and that mstead of zero we find the ex- pression to be equal to e. Then we have (a—c)st+45—d=e, and, subtracting the first of these equa- tions from the second, as in [119], (@—c)S—(@—c)xr=8, or (a—c)(s—2x#) =e. Again, make another substitution s’ for w, and let e’ be the result. Then, as before, (a—o)(s'—axr)=e. Dividing the members of the first of these two equations by those of the last, as in [138], we have (4 —.c) (Ss — &) (a —c)(s' — x) é et and, striking out the common factor amc, solving this equation, as is directed in [111], it becomes successively és—edx=es' —ex, and (e—e)x=es'—e's; whence es'—e's Sy? BAy This expression contains the rule: Make two suppositions for the unknown quantity and note the two false results ; multiply the second supposition by the Jirst result and the first supposition by the second result; subtract the second of these products from the first, and also the second result from the first; divid- ing the first of these differences by the second the quotient is the unknown quantity sought. For example: A doubles the money that B has got, and then B gives A 3/.; when this has been done three times, B finds that he has no money left; How much had he at first? First, suppose that B had 4/. at first. In this case, after the first transaction, he would have 5/., after the second 7/,, and after ARITHMETIC AND ALGEBRA. the third 112. Next, suppose that B had 2/. at first. Then, after the first transaction, he wouldhave 1/7. After the second he would owe A 11., that is, he would have — 17. Since A doubles B’s money every time, the third transaction would be, that A should double B’s debt [see art. 15.]; and then B ought to give A 3/, So that at the end of the third transaction, on this supposition, B owes A 5/., or Bhas— 5/. We thus have Suppositions 4, 2: Results 11, — 5. The product of the second supposition _ by the first result is 22; the product of the first supposition by the second re- sult is —20, the difference of these products is 22 — (— 20) or 42. Again, the difference of the two results is 11 — (— 5) or16. Then, by the rule, thenum- ber sought is the quotient > or 23- This is equivalent to 27. 12s. 6d., which will be found to satisfy the question. The rule of double position is also applicable to solve equations of the form ax+tb=c. But these rules of position are of little use, as all the questions to which they can be applied are much better answer- ed by means of simple equations. Of Arithmetical Progression. 140. Let there be a set, or series, of numbers, such as 4, 9. 10, 13, 165,19, 22, Soon where every one is formed by adding a certain number, in this case 3, to the one before it ; such a set of numbers are said to be in arithmetical progression. The numbers themselves taken collec- tively are also said to constitute an arithmetical series, or progression. The constant number by which every one must be increased, so as to form the next, is called the common difference. The numbers l, 2, 3, 4, 5, 6, &e., for example, make an arithmetical pro- gression in which the common difference is unity. In hke manner, when there is a series of numbers, such as 6, 4, 2, 0, — 2, 4; — 6, — 8, &e., where every one is formed by subtract- ing a certain number, in this case 2, from the preceding, such numbers also are said to be in arithmetical progression. In such a ease the common difference is said to be negative, for every number is formed by adding a negative quantity to the one before it. 141. If abe the first term and 8 the. common difference of any arithmetical progression’ where the common differ- ence is positive, a4,at+b,a+2b,a+3b,&.atnb are the several terms of the progression. In like manner, when the common dif- ference is negative, aa—ba—2b,a—3b, &.,a—nb are the terms. This last series is sim- ply the first with — 6 substituted in it for + 6; so that by giving the pro- per values to a and 4, the first series may be made to stand for any arithme- tical progression, whether the numbers increase or decrease. 142. When we know the first term of an arithmetical progression and the common difference, we can always find any other term of it. ‘The second term is the first with the common difference added to it; the third is the first with twice the common difference added to it, and soon; so thatthe tenthis the first with nine times the common difference added toit; and similarly for any otherterm. This may be stated generally by saying, that then‘ termis equal to the first, with 7 — 1 times the common difference added to it. Thus, if a be the first term, and 0 the common difference, the 2” term of the progression is at+(n—1)8. For example, the 100” term of the pro- gression 5, 11, 17, &c., where the first term is 5, and common difference 6, is 5 +(100 — 1) 6, or 599. As in the last article, this ex- pression may be extended to decreasing progressions by making 6 negative. So that the 20 term of the progression 105, 64, 23, — 18, &c., where — 41 is the common difference, is 105 + (20 —1) (—41), or — 674. 143. When there are two quantities that are the first and last terms of an arithmetical progression,the other terms are said to be so many arithmetical means between them. Thus, 7 and 10 are two arithmetical means between 4 and 13; the progression, when com pleted, being 4, 7, 10, 13. 39 4Q Let it be required to find five arith- metical means between 7 and 25. The five numbers required, together with 7 and 25, are to make an arithmetical pro- gression of seven terms; therefore, by the last article, six times the common difference added to 7 must make 25. 25 — 7, then, or 18, is six times the common. difference, and the common difference therefore is 3. The five means sought are 10, 13, 16, 19, and 22. In the same way we shall find, that if a be one quantity and 7 another, and it is required to find m arithmetical means between them, the common difference of the arithmetical series which these means will form will be l—a \ m+?’ for, just as we divided 25 —7 by 6, which is 5 + 1, in the former case, we shall find that in all cases we are to di- vide 1 —aby m+ 1. This may also be deduced from the expression for the 2‘ term of an arith- metical series [art. 142] in this way: a and / and the m means are to form a se- ries of m + 2 terms; therefore Z is the th m-+2 term of an arithmetical pro- gression whose first term isa; if 5 be the common difference, substituting m + 2 for m for the expression in [142], it becomes ; l=a+(m+ 1), - hence, (972 1-11) bie — ay and, solving this equation, we find as S=a +at+b +qi26 and 8 =at+(n—1)b+a+(n—2) b+a+(n—3) b+ &e. tatb ‘ARITHMETIC AND ALGEBRA. before, nT When there is one arithmetical mean to be found, m is 1, and therefore we divide 7 —aby1+1or2. Onearith- metical mean between 10 and 40 is found by dividing 40 — 10 by 2, and adding the quotient 15 to 10; the sum is 25; and 10, 25, and 40 are in arith- metical progression. 144. Let it be proposed to find the sum of any set of quantities in arithme- tical progression, such as 4, 7, 10, 13, 16,19. Call this sum s; then s=4+74+10+134 16419, or s=19+16+13+104+7+44; taking the terms of the progression in _ an inverted order. Add these two equa- tions together, and they become 2 = 23 + 23 + 23 + 23 + 23 + 23. Now all the terms in the second mem- ber of this equation are the same, and there are six of them. Therefore 2 $=. 6°X-23 =4138; and 138 S= — or 69. 2 By treating in the same way the general expression in [141], we shall find a general expression for its sum, which may be applied to find the sum of any arithmetical progression. Taking mn terms of it, and calling their sum s, we have + &e. +a+(n—2)b+a+(n~1) 8, +a, inverting the order as before, Adding, we have 2s=2a+(n—1)b+2a+(n—-1)b+ &e.+2 a+(n—1)b6+2a+ (n-1) 6. In the second member of this equation we have 2a + (m — 1) 6 repeated n times, since there are n terms in the series. The sum of the terms of this member therefore is the product of 2a+(n—1) 6 by n, whence 2s=n(2a+n—15b), and sere tn= 18) ; fs The expression in [141], the sum of the terms of which we have Just found, became the general expression for a de- creasing arithmetical series, by making 6 negative. Therefore the last expres- s10n for s becomes the sum of a decreas- ing progression, by making 6 negative, In that case ‘yale _n(2a—n—1b) Sg wy Pw and this is the sum we should find for 2 terms of the progression a, a— b,a—2, &e. a — (n — 1) 6, if we treated it in the same way as the increasing progression. The expressions 2a + 2 — 1b, 2a— nm—16 are respectively the first term of an increasing or decreasing progres- sion added to the 2 ; so that it 2 be this 2” term, we have _ nat) * Joule ARITHMETIC AND ALGEBRA, This furnishes us with the general rule for finding the sum of any icreas- ing or decreasing arithmetical progres- sion. Add the first term to the last, multiply this sum by the number of terms, and take half the product. The sum of the progression 1 + 2 + 3+4 &c.up to 7541, is by this rule ae The sum of 40 terms of the progression 23 + 20 +17 + &c. where the 40% 40 (23 — 94) , or 28437111. term is — 94 is or — 1420. 145. The sum of # terms of the pro- gression 1+ 3+ 5+ &c. is by our n(2 +n —1.2) 9 al rule , since the n‘* term of the progression is 1 +” — 1.2. This expression for the sum reduces itself to a 2s 72 or 72. The sum of any number m of the consecutive odd numbers, be- ginning with unity, is therefore the square of 7. Sothat1 + 3 = 2%or4, 1+3+4+5=3?%or9, &.,and1+3+ Bel + &e! + 27 = 142 or 196. 146. It is easy to see that when the first term and common difference are whole numbers, the expression for the sum m.(2a,+ n—1 b) 2 number as it ought; for if 2 be an odd number, 7 — 1 is an even number, will also be a whole and therefore (2a@ + 2— 16) is aneven number. So that whether 7 be an even or an odd number, one of the two num- bers 2, or 2a +n —14, is even, and therefore their product can be always measured by 2. It is plain also, that when a and 6 are whole numbers, and m is measured by 3, the expression for s is measured by 3; for in that case nm (2a + nm — 1 b) ismeasured by 3 [art. n(2a+tn—1b), 51]; and therefore F is measured by 3, [art. 62], since 2 and 3 _are both prime factors of 2 (2 a+n—16). It follows from this, that the sum of any three terms, of any six terms, of any nine terms, &c. im arithmetical pro- gression is measured by 3. Thus, the sum of 12 terms of the series 100, 93, 86, &c, is 738, which is measured by 3. 4] We have seen ‘art. 76] that if any number N be put in the form Ao + A, 10" --+- Ag 10?” +- &e. ; and if Ao + Ay + As + &e. be measured by any number that also measures 10” — 1, then N is measured by the same number. Now 10" — 1 is always measured by 3, [art.67], and we have just seen that if the numbers Ao, Ai, As, &e. be in arithmetical progres- sion, and in number three, or six, or nine, &c., their sum is measured by 3. It follows, that when the digits of a number can be divided into three, or six, or nine, &c. periods, each of. an equal number of digits, if the numbers then expressed by these periods be in arithmetical progression, the number is measured by 3.* In like manner, the sum of any odd number m, of terms of an arithmetical progression is always measured by m, the odd number ; thus, the sum of seven terms is always measured by seven, of nine terms by nine, and so on. There- fore, since 10” — 1 is always measured by 9 [art. 67], if the periods A,, Ag, Ag, &c. be in number 9, or any multiple of 9, and in arithmetical progression, the number is measured by 9. Similarly, since 10? — 1 is measured by 37 ; if the digits of any number can be divided into 37, or 74, or 111, &e. periods of three digits each, and these periods are in arithmetical progression, the number is measured by 37. Of Geometrical Progression, 147. Next, let there be such a set of umbers as 5, 15, 45, 135, 405, 1215, 3645, &e. where every one is formed by multiply- ing the one before it by a certain num- ber, in the present case 3; these numbers are said to be in geometrical progression. The number by which every one is successively multiplied to produce the next, is called the common ratio. This common ratio can always be found by dividing any term in the progression by the one before it. When the common ratio is greater than unity, the terms go on increasing; when it is less than unity, they go on diminishing, as in the series * See Preliminary Treatise, p. 9, in the note, and the examples given there, 42 Pay 16, 8, 4, 2, 1, a? gq? Kes ry e . . ] E in which the common ratio is =< When the terms increase, the series is called an ascending one; when they decrease, it is descending. When the common ratio is negative, the terms are alternately positive and negative, as in the series 11 121 1331 14641 > 10’ 100’ 10007-10000’ tar 11 where the common ratio is — ia! 148. If r be any number positive or negative, the general expression for a geometrical series will be ,47,a77,a7%, a7*..a7". Here a is the first term, a7 the second, ar? the third; and the 2” term is ar"-i, The 7” term of the series 3, 6, 12, &e. is 3 x 28 = 3 x 64, or 192, 149. The intermediate terms of a geometrical progression are said to be so many geometrical means between the first term and the last. Let it be required to find m geometrical means between a and/. The means to be found, together with a and J, are to con- stitute a geometrical series of m+ 2 terms. If 7» be the common ratio, (which as yet is an unknown quantity,) since 7 is the (m + 2)” term of the pro- gression whose first term is a, we have, putting m+ 2 for” in the expression in the last article, l=ar"n, Hence a So that 7 is a number whose (m + 1) power is fA and by [11] this number is a m-+1 l expressed by \/ o We cannot as yet find this number, but supposing known, the means sought are ar, ar, @7.,.ar Since [art. 127] a el par a4 and since a7 is a geometrical mean be- tween a and ar?; it follows that a mean proportional [129] between two quantities is also a geometrical mean between them. It follows from the same article that all the terms of a geo- metrical series are in continued propor- tion. ARITHMETIC AND ALGEBRA. 150. Let it be proposed to find the sum of the terms 8, 24, 72, 216, 648, which form a geometrical progression where the common ratio is 3. Callthe sum s; then $=8+ 24472 + 216 + 648 and 3s = 24 +72 + 216 + 648 + 1944; where the second equation is got by multiplying every term in each member of the first by 3, the common ratio. Now, subtract the first equation from the second, the terms in the second members destroy each other, and there remains 2s= 1944 — 8 = 1936, whence § = 968. 151. Inthe same way the sum of 2 terms of the general geometrical series in art. [148] may be found. Let the sum be s, then s=atartart+ar-+t &e.,, + are? + a7r""1, and rs=artar+aret &e. +ar "2+ ar +ar, just as before. Subtract the first equa- tion from the second; then rs—s=ar—a; whence (r—1)s=ar —-4@, and ar” — a 7 —1 ” r ee If a be 1, these expressions become L+rt+ em + Xe. + nm — | needy and this last is the result we came to in art. [46]. | The expression s = ar—a . a CBives us the following rule for the sum of any number of terms in geometrical pro- gression. Subtract the first term from the last multiplied by the common ratio, and divide the difference by the common ratio diminished by unity ; the quotient is the sum required. For example, the ARITHMETIC AND ALGEBRA. sum of five terms of the series 6, 12, 24, &c. where the last term is 96, and 2x 96 — 6 oe pe? OF 186. So the sum of 8 terms of the series 6, — 18, 54, &c. where the common ratio is — 3, and the 8“ term.is — 13122, is (— 13122).(—3)—6 39366 —6 © ough em SF re a ae — 9840, When the series is a descending one ry andr” are both less than1. In such a case it is better to put the expression for the sum in the form common ratio 2, is or l1—,7" 1—?r S=a ; as in art. [81]. A corresponding change may be made in the arith- metical rule for summing a descending series. _ 152. Let the progression to be summed be = ] 1 1 1 Wa: ag lamas +75 + &e. By the last expression the sum of 2 _ terms of this will be ~ or 1 (<)’ 1 3 n _1-G) -G)" $= Sree aa 1 , 2 2 1 s=2(1-=), or i 1 Ca alien farts. 90, 93]. Now as we goon giving greater and greater values to n, 2”-! becomes greater and greater; and, by making 2 sufficiently great, may be made greater than any quantity that can be named. But as 2”-! becomes = becomes less and less, and may, by making 7 suffi- ciently great, be made less than any quantity that canbe named. It follows, then, that as ” is made greater and _greater, that is, as more and more terms of the series are taken, s differs from 2 by a less and less quantity, and may, by taking a sufficient number of terms, be made to differ from 2 by a quantity less than any that can be greater and greater, 43 named. We conclude, then, that the sum of the series 1 + 5 + ; ay : +&e. if it be continued without limit to the number of terms, is accurately equal to 2; for though we do not show directly that it is so, yet if any one denies it, and states any quantity, however small it be, by which the sum differs from 2, we can show that it does not differ by a quantity so great. For instance, let it be said, that the sum differs from 2 by 1 : : ee 10) oda of unity. Since 2 1024, and since s always differs from 2 by pias if m— 1 = 10, that is, if 2 = 11 sdiffers 1 ; from 2 by oy of unity only. There- fore the sum of 11 terms of the series differs from 2 by a quantity less than the quantity named. The truth of this may be illustrated in this way. AB is a line two inches long, and cut into two equal parts A Cc Dicey ie bas 1S | | Fae at C; CB is cut into two equal parts at I), D B again at E,and so on. Then AC is one inch, C D half an inch, D E a quarter of an inch, and so on; so that AC + CD+DE+EF+&c. is the sum of the series Eben Weert (1 eae git = + &e. ) inches Now we plainly see, that by making subdivisions enough, the sum of these lines may be made to differ from A B, or 2 inches, by a quantity as small as we please, while it can never exceed AB. In these cases we take the sum of a number of terms of the series greater than any number that can be named, that is [art. 117] we suppose n to be infinite. But when is infinitely great, : : , Var 2” is also infinitely great, and oe is in- finitely small, or differs from nothing by a quantity infinitely small, or, in a word, is equal to nothing. 153. Such a series as the one whose sum we have just taken is called an infinite series, because the number of its terms is infinite. Every geometrical 44 series whose terms go on decreasing can be summed to infinity in the same In the expression way. 1 a “i pakd 6-7 a 1-—?r when 7 is less than unity, by making 2 sufficiently great, 7” may be made less than any quantity that can be named. Therefore 1 — 7” may be made to differ from unity by a quantity less than any that can be named; and, as before, we shall have a s= : ~ lar in which we recognise the result in art. [46]. Hence the rule for summing a de- scending geometrical series to infinity: Divide the first term by unity diminished by the common ratio, the quotient is the sum required. The sum of 4 4 - 4 ; ioe Uae Bh iScGss continued to infinity, where the com- Pa le Be mon ratio is —, is by this rule 3] 4 4 ae = 2 or 1. So the sum of 1 4 1 ee fant 4) 5 2 4 8 : : — — — + — to infinity, where the com- 3 9 27. PS. 2 oes mon ratio is — “ay 18 2 2 3 3 fb: 2 2 UE. WS a se oF 3 Of Decimal Fractions. 154. We have seen [art. 74], that we can express all whole numbers by means of the nine digits and zero, by agreeing that every digit when it is written shall | have only one-tenth of the value it would have had if it had held the next place towards the left. _ Let this agree- ment be pursued beyond the units’ place, and let it be understood that when a digit is written next to the right of the units’ digit, its value is one-tenth of what it would have been if it had held the units’ place ; when it is written in the second place to the nght one- ARITHMETIC AND ALGEBRA. hundredth, in the third place one-thou- sandth, and so on. Proceeding from the units’ place towards the left we shall then have tens, hundreds, thou- sands, &c.; and towards: the right, tenths, hundredths, thousandths, &c. For example, 25.763 (where the units’ place is marked by a point placed just after it) will mean 204-5 + J a : : uy 10 100 1900 155. This way of expressing the part of a number that is less than unity is called a decimal fraction. A fraction written in the common form is in dis- tinction called a vulgar, that is, a com- mon fraction. Any vulgar fraction may : Sj be turned into a decimal. Take y for instance, 3 1 3000 — = —-— X — 7; 8 1000 8 or i —— xX 375. 1000 Therefore 3)? 300.7 70>-5 Row 1000 i or 3) 7 5 : 10 100 1000’ and this written as a decimal fraction is .375, where a point is placed to show where the decimal digits begin, and is called the decimal point. In thesame way we find, that ; Ttatiwe 70000 16° T0009. age or 5.4375. Similarly, yl l 100 eS Se 20 100 20 5 ‘ or ——; or «05. Here jzero as placed 1g 100 first on the left, because there are no ] f it j ae tenths of unity in Bn To reduce vulgar fractions to deci- mals, the rule is this: Add zeros to the right of the numerator and divide by the denominator ; make the number of decimal places in the quotient equal to the number of zeros added to the nume- rator, and to make up this number, wf necessary, add zeros to the left of the quotient. A zero placed to the right of a decimal has no effect; .2 and .20, for instance, are respectively the same ARITHMETIC AND ALGEBRA, ' 45 2 - - "20 Pash —., both of whi i 7 and i both of which are equi valent to E. 5 156. If by this rule we try to reduce E fo a decimal fraction, the division will never come to an end, and we find for answer .333 &c., without limit. We shall find by the rule in art. [150], that the sum of the infinite geometrical se- ries Sth dail Daas On 10 ' 100 * 1000 ; which is equivalent to the decimal .333 It is easy on other grounds to see the reason of this result. When we multiply the numerator of a fraction by a number and divide the product by the denominator, it is necessary, in or- der that the division may leave no re- mainder, that the numerator be multi- plied by some number that makes it a multiple of the denominator. Now, in our process, we multiply the numerator by some power of 10, and no power of 10 is measured by 3; therefore we may divide the product by 3 for ever without coming to an end. The only prime factors [art. 61] of any power of 10 are 2 and 5; so that when a fraction is in its lowest terms [79], if there be any prime factor be- sides 2 and 5 in its denominator, it eannot be reduced into a decimal that will terminate. In that case, the nume- rator contains none of the prime factors of the denominator, and no power of 10 contains them all, therefore the pro- duct of the numerator by no power of 10 is measured by the denominator [61 |. 1 &e., is ae ae Thus, = gives the decimal .555 &c., ee . and 59 Bives .6818181, &c. Decimals such as these are called repeating de- cimals ; those in which the repeating part consists of more than one digit are sometimes also called circulating deci- mals. .333 &e. is written .3, so .68181 &e. is written .681, and .531531 &c. is written .531; the points being placed so as to mark out the repeating part. 157. When a proper fraction is in its lowest terms, it can therefore always be reduced to a terminating decimal if there be no prime factors in its deno- minator other than 2 and 5, and in that case only. The number of digits in the decimal to which it can be so reduced will be the number indicating the power of that one of the two factors 2 and 5, which is found in the highest power in : Ha the denominator. For example, ie 8 reduced to .3125 with four digits, be- Cause 16 is the fourth power of 2, So Lae ; 59 8 reduced to .38, with two digits, because 5 is found in its second power in 50, while 2 is found only in its first. Let $ oti as be the fraction, where a does not contain 2 or 5; and let m be greater than 2. Ifwe multiply a by.10”, that is, by 2”. 5™, it will be measured by 2”.5”; while if it be multiplied by any lower power of 10, as 10"~1, it will not be measured by 2”, and therefore not by 2”.5". Now,when we multiply by 10" we add m zeros, and by our rule the number of decimal places is equal to the number of zeros added. 158. When the fraction is in its lowest terms, and the denominator is prime to 10, we have seen that it will not pro- duce a terminating decimal. It will always give a repeating decimal, which will begin to repeat from its first digit, The reason is plain. yi : G Thus Ta produces .53846 1, and = pro- duces 714285, as follows. Let > be a proper fraction where 6 is The reason of this is prime to 10. In turning it into a deci- mal we divide 10 @ by 6; the whole part of the quotient, which we may eall d,, is the first digit of the decimal ; let the remainder be 7,. We divide 107, by & for the second digit d,, and have a re- mainder 7,. Proceeding thus we find digits d,, ds, d,, d,, &c., and remainders 11,72,13, 74, &C., Corresponding. Now suppose that after a certain number of partial divisions we fall on a remainder, rz for instance, the same as one we have had before. The remainders that succeed 72, the second time, must be the same as those which succeeded it » at first; for the circumstances to pro- duce them are the same. In both cases they will be 7,, 7,, &c. But the remainder preceding 7? in both cases will also be the same, viz. 7,. 46 Suppose, for the sake of argument, that in the second case it may be different, and call it s. Then we have 107, =d,b6+ 7, in the first instance, and in the second l0s=qb+n7,. Where g is some whole number less than 10, and different from d,, else 7, and s would be the same. Subtracting the second of these equations from the first 10 (7,— s) = (d, — q) 6, whence 10 (1 —_ S$) eT —- d, — q. Now all the remainders 71, 72, s, 7, &c. are less than 4, and therefore 7, — s is less than 6; therefore 5 cannot mea- sure 7, —s; and therefore since 10 is prime to 4, 6 cannot measure 10 (7, —s), which on our supposition it does. It thus leads to an absurdity to suppose that 7, and s can be different, and there- fore they are the same. In the same way it may be shown, that the remainder preceding 7; the second time it occurs, is a. It follows, that if among the re- mainders any two are the same, all the remainders form periods exactly similar to each other, the second period beginning with a. But among the re- mainders some two must be the same, for they are all less than 4, and there- fore there can be but b—1 different ones, while their number is infinite. Now when we come to a at the beginning of the second period of remainders, the digits furnished will be d,, d,, ds, &C., just as when we began from a at the beginning of the first period. There- fore the circulating part of the decimal will begin by repeating the first digit of the decimal. The number of digits in the repeating part cannot be more than 159. When the fraction is in its lowest terms, and its denominator is not prime to 10, while it contains other factors besides 2 and 5, the decimal will, as we have seen, repeat, but will not begin from its first digit. If the m*t be the highest power of 2 or 5 in the denomi- nator it will begin to repeat after the aoe 6 Med A mth digit, Thus, = is reduced to .83, which repeats after the first digit; be- cause 6 contains 2 in the first power. 23 So — 120 digit, because 120 contains the third or-.1916 repeats after the third ARITHMETIC AND ALGEBRA, power of 2, and only the first power of 5. The reader will have no difficulty in deducing this from the principles just laid down, 160. To reduce any terminating deci- mal to a vulgar fraction: Write the decimal as a whole number for the nu- merator, and unity followed by as many zeros as there are digits in the decimal for the denominator; reduce the frac- tion so formed into lower terms tf pos- sible. The decimal .725, for instance, 7 2 5 1s ivé — — + —,or to is equivalent es + vy + ane 700 20 4 5 Soe 72 get 1000 1000 § 1000’ 1000’ 40° 161. To reduce a repeating decimal to a vulgar fraction, is to take the sum of an infinite geometrical progression, [156]. Thus .231 is equivalent to ae | Zan 1000 1000000 1000000000 Am 1 where the common ratio is Fat The sum is by [153] es 231 231 1000 _ 1000 231. 77 : qrctiwat!f99e BF 998's0n1 S835 ~~ 1000 1000 ‘ If P be the repeating period, and con- tain m digits, the value of the decimal is a z H + &e. To" * fom T Tom sane ea 3 The common ratio is ries and the sum is snes cage [art.153]. Observe that 10"—1 is a number composed of m nines, Similarly .1737, which does not repeat from its first digit, is equivalent to 17 37 37 100 10000 1000000 37 2 es ee Fe & e 100000000 Biss that is, to 17 100 a7 4 4 hi 1 i 1 zy 100(100 © 10000 1000000 rm or 17 37 1 100 ' 100 * 99° or 17 37. 86 100 ' 9900 495 ARITHMETIC AND ALGEBRA. From these examples we collect the following rule: When the decimal re- peats from its first digit, the vulgar fraction equivalent to it has for tts numerator the repeating period, and for ats denominator as many nines as there are digits in the repeating period ; when it does not repeat from its first digit, the decimal fraction ts equal to the sum of two vulgar fractions, the first of which is the value of the part not repeating considered as a terminating decimal, the second has for its numerator the repeating period, and for its denominator as many nines as there are digits in the repeating period. followed by as many zeros as there are.digits in the part not repeating. 162. To add or subtract terminating decimals, or mixed numbers containing them: Write them one under the other, observing to have the decimal points all in the same column; then proceed as in whole numbers. When they are written in this way, every column contains all the digits of some one denomination. The column to the left of the decimal points, for instance, contains all the units, the one to the right all the tenths, and so on. This being the case, the reasons in arts. [20] and [21] apply equally here. 163. To take the product of terminat- ing decimals or mixed numbers con- taining them: Multiply them together as uf they were whole numbers, and point off from the product as many decimal places as there are in the mul- tiplier and multiplicand together. If there are not so many digits in the product add zeros to the left. Let 5.75 and 12.731, for instance, be the ; : 75 575 ‘ Ip 5 = — or —. two decimals, 5.7 5 + re 100 12754 , 632 127.54 = 6.32 = — — —_ tical 100 * 100 12754 , 632) 1S TRA eGR re ee ee ae in 1000 * 10 2754 | 632 ore eudigentei yaa 10 1000 Here we have three cases in which the divisor and dividend consist of the same digits. In the first the number of deci- mal places is the same in both, and the quotient we find to be the same as if they were whole numbers. In -the second there are three decimal places in the dividend, and one in the divisor; the quotient, then, is found by adding two zeros to the divisor. In the third case, Where the dividend has one decimal 47 12731 Similarly, 12.731 = ———_, _ Th imilarly, 12.7 Ti Therefore 575 12731 575 12.73) = —— 45 eee , 100+ 1000’ or 575 X 12731 100009 Taking this product it will be found to _ 7320325 20325 sg ——_ or eee 100000 00000 73.20325. We point off five places, be-~ cause there is 100000 in the denomina- tor of the product; that is, because there was 1000 in the denominator of one factor and 100 in that of the other ; or because the original factors had re- spectively two and three decimal places, ee 16 23 Similarly .0016 x .023 = aaa 4 10000 1000 368 a AE ag ae .0000368. 10000000 oe When the multiplier or the multipli- cand is a whole number, we point off as many decimal places as the other factor contains. Thus, 18 x 2.75=49.50 or 49.5, 164. To divide one decimal fraction by another: Make the number of digits after the decimal point equal in the divisor and dividend by adding zeros to the right of one of them, tf necessary ; then divide as uf they were whole num- bers, the quotient is the quotient re- quired; the remainder gives a vulgar Sraction which may be turned into a decimal by adding zeros to it, and dividing by the divisor, This rule may be deduced from the following example : 12754 100 12754 TET U0. capes ea eee 12754 10 «12764 ~ $1000» 632... 63200 ” 12754 1000 1275400 Ride Gian |, 62a 7: place and the divisor three, we add two zeros to the dividend. When there are more decimal places in the dividend than in the divisor, this rule amounts to dividing the dividend by the divisor, and pointing off as many decimal places in the whole part of the quotient as the number of decimal places in the dividend exceeds that in the divisor, or if there are not so many in the quotient, adding the requisite 48 number of zeros to the left. This is the form in which the rule is often stated, and may, perhaps, be found the more convenient. Following our rule, when one of the numbers is an integer, before dividing we must add to it as many zeros as there are decimal places in the other. 165. If N be any number, and if a mixed number, with its fractional part expressed as a terminating decimal, have the same digits as N, and if there be » decimal digits, the mixed number is 1731 ae OE: N \ equal to 14 Thus, 17.31 = aD ates Similarly, if all the digits be decimal and there be » of them, the value is aa _ On the other hand, any , Ny. : fraction such as pn equivalent to a number consisting of the same digits as N, 2 of which are decimal. Thus 7243 100 From this we may easily deduce the ules of multiplication and division = 72.43. N given above. The product of TiC and ASS NN! ——, is ] 0” n and n! of its digits made decimal. But N N’ — and —. 10” 10” bers N and N’ witn 2 and m! decimal digits. Therefore to take the product of two numbers with » and n’ decimal places respectively, we point n + n’ digits from their product considered as whole numbers. or the number N N’ with 1o7+e”’ are respectively the num- Again the quotient of ~ divided by See pate Cal — is —.—_, 10” Vo Sy greater than ’, that is, when the number of decimal places in N exceeds that in N’, this becomes [art.92]. When n is ng Theref N’/.10%-””" ' ere ore to N’ we must add n— 7! zeros, and n — n' is the excess of the number of decimal places in N over that of those in N’, Similarly, when w’ is greater than ARITHMETIC AND ALGEBRA. N..104(~t N’/ Showing that in that case we must add to N n! — n zeros. 166. With respect to repeating deci- mals, if perfect accuracy be necessary, they must in most cases be reduced to vulgar fractions before they are added, subtracted, multiplied, or divided. In almost all the applications of decimals, however, an approach to accuracy is sufficient, and this is attained by carry- ing the decimal only to a moderate number of digits, and omitting the rest. If, in converting a vulgar fraction into a decimal, we stop after the third digit, for instance, adding unity to that digit, if the next be 5 or upwards, it does not differ from its exact value by more than one five-thousandth part of the unit employed. Thus .172 differs from 172437 by .0004372, which is less than .0005. Similarly, .983 differs from -98276 by .0002317, which is also less than .0005. Decimals are most frequently used to make calculations on numbers that have been obtained by observations of some kind, by measuring, for instance, or weighing; and it is very seldom in- deed that the accuracy of these obser- vations can be relied on to within one five-thousandth part of the unit em- ployed. Now if we cannot rely on the measurement beyond three decimal places, it is needless to carry the result derived from it any farther. In all operations with decimals, then, whether terminating or repeating, we may usually stop at the third or fourth place, and need very rarely go beyond the fifth or sixth. We may, however, attain any degree of exactness that may be re- quired, by carrying the decimal far enough. Though the quantity thus neglected be very small, it is not less than any quantity that can be named. The ac- curacy of the result is therefore very different from that of such results as the one in art. [152], where the error was proved to be less than any that could be named. ‘There the result was exact, here it only approaches to exact- ness. 167. The operations of multiplying and dividing decimals may be much - shortened when this degree only of ac- curacy is required. For example, to multiply 2.753 by 2.3i, carrying the pro- duct to three places of decimals, n, this quotient becomes ARITHMETIC AND ALGEBRA. 2.753 2.753 2.753 2.313 2.313 2.313 8259 5506 55U6 2753 8259 826 8259 2753 28 5506 8259 8 6.367689 6.367689 6.368 The first multiplication is in’ the usual way. The second is the same, but with this difference, that instead of multiplying first by the right hand digit of the multiplier, and shifting every partial product one step to the left, we multiply first by 2 the left hand digit é of the multiplier, and shift every partial product one step to the right. The order of the partial products is thus in- verted, as will be seen by comparing this multiplication with the first; but, except in the order of these partial pro- 49 ducts, the two processes aré the same. In the third, which is the shortened form, we proceed as in the second, only we never write any digit to the right of the column that furnishes the third decimal place in the product, and we always add unity to this extreme digit when the next appears to be 5 or up- wards. A trial or two will show how much more convenient it is to begin the shortened process from the left than from the right of the mul- tiplier. The exact result would be found to be 6.36805, which differs from the approximate result by of 19800 unity. Again, to divide 49.782 by 2.7167 carrying the quotient to three places of decimals : 2.7167)49.7820(18.3244 2.7167)4 9.7820(18,324 27167 226150 217336 88140 8150)\1 90 34 560 668 8920 8668 The first operation is as the rule directs. The second is the shortened form, in which, instead of adding a zero to every remainder, we cut off one more digit from every successive product, in- creasing the last digit by unity when the next neglected one is 5 or upwards. On comparing the examples, the effects will be seen to be alike. These are examples of the shortened methods. The reader will have no difficulty in extending the same principles to other cases. 168. To reduce a number, whole or decimal, of a lower denomination [see art. 98, &c.] to a decimal of a higher: Divide it by the number of times that the unit of the higher denomination contains that of the lower. For exam- ey : 1 ple, 6.35 shillings = 6.35 x mT of a 6.3! Pay 6M £.3475. > So 29 to reduce 6s. 43d. pound ; pound, that is £ to a decimal of a 27167 226150 217336 8814 8150 664 543 121 109 —___ 12 3 zt. => «OG: 4.75d. = awe, or .39583s. ° . 8 3 e 6.39583s. = a =o a or £.3197916. And therefore this last is the decimal sought. 169. To reduce a fraction or a mixed number expressed as a decimal of a higher denomination into a number of a lower denomination: Multiply it by the number of times that the unit of the higher denomination contains that of the lower. Yhus, 5.27 feet = 5.27 x 12 inches, or 63.24 inches. So to reduce £.1368 to shillings, pence, and far- things; £.1368 = 20 x .1368, or 2.7368. 4008. = 195 .7386, or 8.832d. .832d. = 4 x .832, or 3.328 fars. The answer therefore is 2s. 8d. 3.328 far- things. E 50 170. Only a small proportion of num- bers have no other prime factors but 2 and 5, and therefore only a small pro- ortion of vulgar fractions can be re- duced to terminating decimals. If twelve were the base of our scale of: notation [73] instead of ten, all those fractions whose denominators have ‘no prime factors but 2 and 3, or one of these numbers, would terminate if pre- sented in that scale in a form similar to ; : 1 decimal fractions. For imstance, 5 1 would become .6, 5 would become 44, ars 1 — would become .2, —— would be- 6 pee come .054, and soon. ‘There are more numbers that have 2 and 3 only for their prime factors, than there are that have 2 and 5 only. Of the numbers be- tween 1 and 100, for example, 18: belong to the first class, and only 13 to the second. As this way of expressing fractions is very convenient, and as terminating decimals are much more manageable than repeating ones, twelve, to the extent of this reason, would be a better base for a scale of notation than ten. Of the Square and Cube Roots, and of Surds. 171. The square root of any number ais a number which, when multiplied by itself, produces a [art. 11]. When the square root of a number is an in- teger, the number is called a square number. Thus 4 is a square number, because 2 multiplied by itself produces 4. So144 is a square number, for it is produced by the multiplication of 12 by itself. 172. The square root of a whole number that is not square, cannot be expressed by means of any fractional part of a whole number. Thus, if a be a number not square, its square root P cannot be expressed in the form 7 + =, where 7 is a whole number, and P an irreducible fraction. For if it could, we should have (+BY ARITHMETIC AND ALGEBRA. or, multiplying 7 + a by itself, [see art. 90] and, consequently, 2 P =q—72—2r aS am Multiply both members of this equation by q, and it becomes pe —=aq—Tq—2rp. q Now p is prime to q, therefore p® is prime to q, and therefore it is not pos- 2 sible that = ean be equal to a whole number, as we have found it to be. It follows, that it is not possible that ,/a can be expressed in the form 7 + a This is-a remarkable property of such square roots. It amounts to this, that it is impossible to find any num- ber, however small it be, that will at the same time measure unity and ,/a, when aisnotasquarenumber. Ifit were pos- sible to find a number = that measured both, then ,/a might be presented in the form 7 + 4 , and that we have seen is impossible. This is usually expressed by saying that ,/a is imcommensurable with unity, when a@ is not a square number. Thus 4/2, 4/3, 4/15, &c. are all incommensurable with unity; there is no fraction, vulgar or decimal, that is exactly equivalent to them, or to any similar numbers. 173. In the same way, a is called a cube number when its cube root [art. 11] is a whole number. Thus 27 and 125, the cube roots of which are 3 and 5 respectively, are cube numbers. As before, it may be shown, that where a is not a cube number, its. cube root is incommensurable with unity. So if +,/a [art. 11] is not a whole number, the fourth root of a is in- commensurable with unity. All such incommensurable numbers as A/ 2, 84/5,.4+./9, &c. are called irrational numbers, or surds; while numbers commensurable with unity are called rational numbers. ‘There are. many ARITHMETIC AND ALGEBRA. numbers besides surds that are incom- mensurable with unity. 174. We now proceed to explain the common methods of finding the square and cube roots of numbers, and in doing so we shall have occasion to refer to the following table of the squares and cubes of the first nine numbers. Number, Square. Cube. ble 1 1 2 4 8 3 7 27 4 16 64 5 25 125 6 36 216 7 49 343 8 64 512 9 81 729 The square of any digit followed by any number of zeros, is the square of that digit in the table, followed by twice the number of zeros; and the cube is the cube of the digit in the table, followed by three times the number of zeros. - Thus, the square of 700 is 490000, and the cube of 90 is 729000. 175. One way of finding the square root of a square number, would be to guess some number that might be near it, and try whether its square were greater or less than the number proposed. We might then correct the number guessed and try again, and so continually ap- proach the square root sought till at last we should find it. The common way _ of finding the square root of a number is, like this, a succession of trials, only _ much less tedious; for we are able to find the digits in the square root one by one, and to correct them without the ‘trouble of squaring at every step. The square of every number between _ 1 and 10 is between 1 and 100, and has _ one or two digits. The square of every - number between 10 and 100 is between - digits. ~ number between 100 and 1000 has five 100 and 10000, and has three or four Similarly, the square of every or six digits, and so on. On the other hand, every square number consisting of one or two digits has one digit in its square root, every square number of three or four digits has two in its square root, every square number of five or six digits three, and so on. It follows, that when a square number is proposed, if we divide it into periods con- sisting of two digits each, except the last, which will consist of two digits or one as the case may be, there will be as many of these periods as there are digits in the 51 square root of the number. This divi- sion into periods, or pointing, as it is called, should begin from the right of the number, as will appear afterwards. 176. Let 4624 bea number of which the square root is wanted. In the first place, by,pointing it thus, 4624, it ap- pears that the square root has two digits. Next, we observe by the table that the square of 70 is 4900, while the square of 60 is 3600, the one greater and the other less than 4624. The square root sought is thus between 60 and 70. Call it 60 + 0b, where 6 is necessarily less than 10. Then (69 + 5)2 must be equal to 4624, or, multiplying 60 + 6 by itself, 602+ 2x 606+ & = 4624, From each member take away 60, or 3600; then, altering the first member a little, there remains ; b(2 x 60 + b) = 1024, whence : 1024 1024 Sa, OY ————_, 2x 60+6 120 +6 Now 6 is less than 10, and therefore so much less than 120, that if we neglect it in the denominator, the result, which . 1024, : iS Fag? will not be very different from 1024 nett 799 38 pretty nearly 8, so that we may try 8 for 6. If we square 60 + 8, or 68, we shall find the product to be 4624, and there- fore 68 is the square root sought. This example teaches the principle on which the rule for extracting the square root is founded. We first find the first part of the root. We then subtract the square of the part found, and the appearance of the remainder. enables us to determine the second. part. We then subtract the square of the first and second parts, and the appearance of the remainder enables us. to determine the third, and so on. 177. Suppose that we find a to be the first part of the root, and that, sub- tracting a2, we find a remainder from which it appears that 6 will be the second. To ascertain this we must as before subtract (a + 6)? from the num- ber proposed, : But (a+ b)2 = a2? +2ab+ &, the true value of 06. or (a+ bf =@2+bQa+d), | and therefore if we subtract 6 (2a + 6) from the first remainder, it will be the EB 2 52 same thing as subtracting (a -+ 5)? from the original number. In lke manner, when we have determined (a + 6), sup- posing that ¢ is the third part, we must subtract (a + 6 + c)? from the original number. But (atb+o2= $4Get0D) ee), and considering a+ 6 as one quantity, we have (a+b+ c= (at 6b) 21a FO) Cre, or (a+ b+ 0)? = (a+ 6) +e}2(at b) + Cae as before ; just as before then, to sub- tract (a + 6 + c)? from the original number, is equivalent to subtracting Cc {2 (a+ 6) + ch from the second remainder. Observe how these properties shorten the following example: To extract the square root of 322624. 322624 (500+ 60+ 8 Subtract 250000 = 5002 1000 + 60) 72624 Subtract 63600 = 60 (1000 + 60) 1120 + 8) 9024 Subtract 9024 = 8 (1120 + 8) We first find, by pointing, that there are three digits in the square root. It next appears by the table, that the root is between 500 and 600. We subtract 5002, and dividing the remainder as be- fore by 2 x 500 or 1000, we should first find, as in last article, that 70 was the next part of the root. But we find that 70(2.500 + 70), or 74900, is greater than 72624, and therefore 5702 must be greater than 322624, as has been shown. We then try 60 for the second part. We subtract 60(2 x 500 + 60) which, with 500° already subtracted, makes, as we have seen, 5602 in all. To find a number to try for the third part of the root, we divide the last remainder as before by 2 x 560, and find the quotient about 8. We then subtract 8 (2 x 560 + 8) equivalent to c [2(a+4)+e!} above, and making in all 568? subtracted, and as we then find that nothing re- mains, 568 must be the number sought. Omitting useless digits, this process may be put thus: 322624 (568 25 106) 726 636 1128) 9024 9024 —_—_— ARITHMETIC AND ALGEBRA.’ And this agrees with the usual rule, which is as follows: Potnt every se- cond digit beginning with the units ; Jind by the table the greatest digit whose square is contained in the left hand period, and write tt as the first digit in the root; subtract its square Srom the left hand period, and bring down the second period to the remainder to form a dividend ; to the left of this dividend write twice the digit last found as animperfect divisor ; divide the divi- dend, omitting its last digit, by the im- perfect divisor, andwrite the single digit, which is the whole part of the quotient, as the second digit in the root, and also to the right of the imperfect divisor to JSorm the perfect divisor ; multiply this divisor as it now stands by this digit, uf the product be greater than the divi- dend the digit is too great, and a smaller one must be taken; when the digit is not too great, subtract the pro- duct last mentioned from the dividend, if the remainder be greater than twice the number composed of the two digits written in the root, the digit last found 7s too small, and a greater one must be taken; if the digit be not too small, to theremainder last mentioned bring down the third period to form anew dividend, and to the perfect divisor, as it now stands, add the digit of the root last Sound to form a new imperfect divisor, after which proceed as before. The reason why the second digit is too small, when, after the product of it by the perfect divisor is subtracted from the dividend, the remainder is greater than twice the part of the root found, may be explained thus. Let M be the value of the two first periods of the number, and P that of the part of the root found. The remainder in ques- tion is, as we have seen, the difference M—F*. Now if M — F2 be greater than 2P, it follows that M is greater than P2-+ 2 P, and since M and P are whole numbers, M must either be greater than P? + 2P + 1, or equal to it, But P?+2P+1=(P+1)2; there- fore M is either equal to or greater than (P+ 1% It follows that P +1 or some larger integer, and not. P, ought to be the first part of the root, so that the second digit is too small. The same .reasoning, of course, applies to any subsequent digit. ; 178. With respect to the square root of a number not square; observe that when we multiply any number by 100, we make its square root ten times as ARITHMETIC AND ALGEBRA. preat as it was before. The square root of 100a@ is 10,/a, or ten times the square root of a; since if we multiply 10,/a by itself, the product is 10 x 10 x Jax Ja, or 100 a. So, when we multiply any number by 10000, or by 1000000, we make its square root 100 _ times, or 1000 times as great as it was before. Now, suppose we have to find the square root of 129. We find by the rule that 11 is the first part of it, but there is a remainder over after 11? is subtracted. If to this remainder we add two zeros, and go on another step, we shall proceed as if 12900 had been the number proposed, and we shall find that 113 is the whole part of its square root. But the square root of 12900 is, as we have seen, ten times as great as that of 129, and therefore or 11.3, is anearer approach than be- fore to the square root of 129. So if we again add two zeros to the remainder when (113)? is subtracted, we shall find 5 to be the next digit, showing that 1135 is the whole number next below the square root of 1290000. Therefore UNS? | or 11.35, is a still ne 1007? OF 11.35, is a still nearer ap- proach to the square root of 129. If we add two more zeros we find the next digit to be 7, and the next in the same way to be 3, so that 11.3573 is a still nearer approach to the square root of 129. Proceeding in this way we can eome as near to the square root of 129 as we please, though, for the reason in arf. [172], it can never be expressed exactly in decimals. We must make the following addi- tion to our rule to meet this case: When there is a remainder over, after the last period is brought down, add two zeros to it, and proceed, as before, placing the decimal point before the digit next found. 179. The rule for finding the square root of decimals is on the same prin- ciple. The square root of 21.76 is one tenth of that of 100 x 21.76, or of 2176. So the square root of 97.968 is one hundredth part of that of 10000 x 97.968, or of 979680. Simi- larly, the square root of .034 is one hundredth part of that of 10000 x .034, or of 340. The rule then will be this: Make the number of digits after the 53 decimal place even, by adding a zero to the right if necessary ; proceed as dt- rected for whole numbers, making half as many decimal places in the root as there are in the number proposed as tt now stands, For example, to extract the square root of .07361: there are five digits, therefore by the rule we must add a zero to the right so as to make if .073610. We then point as if it were a whole number, thus 076110. There will be three decimal places in the root furnished by the three periods in the decimal as it now stands. 180. Asin the case of the square root, and on the same principles, we can know at once how many digits there are in the cube root of any cube number. The cube of every number between one and ten has one, two, or three digits, of every number between ten and one hun- dred, four, five, or six digits, and so on. So, on the other hand, if a cube number consists of one, two, or three digits, its cube root has but one. If it consists of four, five, or six digits, its cube root has but two. If it consists of seven, eight, or nine digits, its cube root has but three, and soon. So that if any cube num- ber be divided into periods of three digits each, except the last, which will contain one, two, or three, as the case may be, the number of these periods will be the number of digits in its cube root. This dividing into periods, as in finding the square root, should begin from the right of the number. 181. We find the cube root of a number part by part. When we have found the first part we subtract its cube, and from the appearance of the remain- der we determine the second. We then subtract the cube of the first and second, and from the appearance of the remain- der determine the third, and so on. Now, let N be the cube number pro- posed. Suppose that we find that a is the first part of its cube root, we subtract a?; let the remainder be R. Call the rest of the cube root 2, then (a + x} must be equal toN. (a4 + 2), is equal to (a+zxv) x (a+ x}, or to (a+ x) (a2?+ 2axr-+ 2%), and multiplying, this becomes @+3a@r+3aa2t+ x. Therefore e+ 3ae+3ar+H=N, 54 and SU rr SO ze whence N—@=R; R = 3a@+3an+2° 3 But x is always less than a, and there- fore 3 a2 is usually much greater than 3aa +22, As an approach then to 8, the next part of the root, we may take a number near -—.. [See art. 176.] When we have found 6, we subtract (a + b)%, and ealling the remainder R’ we find the next part by taking a num- R/ 3 (a + 6)? As will be seen, the operation is much shortened in consequence of the two following properties : ber near ARITHMETIC AND ALGEBRA, First, (a+ 68 =a+ 30a°b+ 8ab2 + 8, as before; or (a + bP = a+ b(3 a8 + 3ab + BY; or, finally, (a+ b= a +b)3a@+b(8a+b)}. Second, since (a+ bP =a + 2ab+ 2b, then 3(a+ b2=3a+6a64+3 &, or 3(a+ b6)?%= 3a2+ 56(3a+ 5b) +6b@8a+t+B5) +.B. These properties may be extended to the case of a+ b+ ¢, or further, as in art. [177]. Let us now proceed to find the cube root of the number 98611128. 3 a=1200 b=! *60 3a+b=1260 2b= 120 b(3at+b)= 75 600 3 (a+ 6) =1380 380@+6 (3at+6)=555 600 c= 2 &&= 3600 8at+6+¢=1382 Pointing the number, it appears that there are three digits in its cube root. Next, the root appears to be between 400 and 500, since by the table the cubes of these numbers are respectively 64000000 and 125000000. We sub- tract the cube of 400, and find the re- mainder, which we have called R. To the left of this remainder we set down 3 x 4002, or 480000, as the amperfect divisor, and to the left of that 3 x 400, or 1200. Dividing R by the imperfect divisor, the quotient is about 60, which, as we have seen, will be a near approach to the next part of the root. We add 60 to 1200 on the left, and multiply the sum 1260 by 60. We then write the product 75600 under the imperfect divisor, and add these numbers for a perfect divisor. We multiply the sum 555 600 by 60, and write the product 33 336 000 under the first remainder R and subtract. Now we shall find from its composi- tion, that the number last subtracted is 60 | 3.4008 + 60(3 x 400 + 60) } a b ¢ 98 611 128 | 400+60+2 64 000 000=ae ; 3 a2=480 000|\34 611 L28=R 33 336 000=5 {3 a&+h(3a+b)} 3 (a+ b)2=634 800 e{3(atb)t+ce!= 2764 637.564! a 275 128= RY 1 275 128=c}3.a+b'+c(3.a+6 +¢ and this with 400% already subtracted, makes up (400 + 60)3, by the first pro- perty. The remainder 1275128 is there- fore what we have denoted above by R’. For a new imperfect divisor we write 60? under the last perfect divisor. We then add 602, and the two numbers above it together. The sum from its composition is 3 x 4002 + 60(3 x 400 + 60) + 60 (3 x 400 + 60) + 608, or, by the second property, 3 x 460% Lastly, to the number on the left we add 2x60, or 120, the sum 1380 is 3 x 460. Dividing by the new imperfect divi- sor, we find that 2 is the next digit of the root, with which we proceed as be- fore. We add 2 to 1380, multiply the sum by 2, and add the product to the imperfect divisor for a new perfect divi- sor. We then multiply this perfect divisor by 2. When we have sub- tracted the product, we have, as ‘has been already shown, subtracted 4623, “ARITHMETIC AND ALGEBRA. 55 and as there is no remainder, 462 must be the root sought. This operation may be shortened, as in the next example, by omitting useless digits. To extract the cube root of 625245708151. 245 625 245 708 151 | 8551 10 512 2555 19200| 113 245 10 Leos 25651 20425| 102 125 25 2167500|/11 120 708 12775 2180275|10 901 375 25 219307500 | 219 333 151 25651 21933a101 1.219 saad. 15), 182. When the product of the per- fect divisor, and the digit assumed is greater than the preceding dividend, that digit is of course too great, and a smaller one is to be taken. Again, if the remainder, when this product is subtracted from the dividend, be greater than 3 P? + 3 P, where P is the part of the root found, the digit last assumed is too small, and a greater must be taken. For let M be the value of the periods already brought down, then the re- mainder in question is, as we have seen, M—P:. If this be greater than 3 p2 + 3P, it must follow that M is greater than P?+3P? + 3 P, and that M is either equal to, or greater than pa+3P24+3P+i. Now this last expression will be found equal to (P + 1), and if M be equal to or greater than cP. +.1)%) P+ 1, or some larger integer, and not P, ought to be the part of the root found, and there- fore the last digit must be increased by unity. When there is a remainder after all the periods are brought down, the num- ber is not a perfect cube. But, for the same reason as in extracting the square root fart. 178], if we add three zeros to this remainder and proceed, the digit next found will be the first decimal digit in the root, and by continuing to add zeros three at a time, we may ap- proach to the cube root as nearly as we please. 183. The rule may be ‘stated thus: Point every third digit in the number proposed, beginning with the units’ ; find by the table the greatest digit (a) whose cube is contained in the left hand period, and write tt as the first in the root; subtract a3 from the first period, and to the right of the remainder write the second period to form a dividend ; to the left of this write 3 a?, to which an- nex two zeros to form an emperfect divisor ; still farther to the left write 3a; find a second digit (b) by dividing the dividend by the imperfect divisor, and annex tt to the right of 3a in the column farthest to the left ; multiply the number in this column as it now stands by b, and write the product under the imperfect divisor for a correction ; add the imperfect divisor to the correc- tion, and write their sum, which ws the perfect divisor, under them ; multiply the perfect divisor by b, and subtract the pro- duct from the dividend, and to the right of the remainder annex the next pe- riod; for a new imperfect divisor write b? under the last perfect divisor, and take the sum of b? and the two numbers above it; then add 2b to the lowest number in the column farthest to the left, and proceed as before; the several digits so found are the digits of the root. When there is a remainder at last, add three zeros to it and proceed, the digits afterwards found are decimat parts of the root. 184, When part of the number pro- posed is decimal, the rule is founded on the same reasons as for the correspond- ing case in the square root, [art. 17 9.| It is this, Make the number of digits after the decimal point three, or some multiple of three, by adding one or two zeros to the right if necessary; then proceed as is directed for whole num- bers, placing the decimal point before that digit in the root which is first 56 ARITHMETIC AND ALGEBRA. found after taking down the first decimal period in the number pro- posed. For example, to extract the cube root of 21.7698, that is of 21.769800. This root is [art. 179] one hundredth part of the cube root of 21769800. Point- ing the number thus, 21769800, and extracting its cube root, we must mark off two decimal digits from the whole part of the result, and this amounts to what is directed by the rule. Under these rules for pointing de- cimals before extracting the square and cube roots, a point always falls on the units’ digit, and consequently on every second, or on eyery third digit to the right and left of it. 185. The operations for finding the Square and cube roots of numbers ad- mit of being shortened in a way similar to that adopted in [art. 167]. “For in- Stance, let it be proposed to find the Square root of 329.i to five places of decimals. Adding a zero, as directed by the rule, and pointing, we have 329.16. 275 10 2854 24300 8 1375 28623 6 25 286296 _ 2707500 11416 2718916 16 273034800 25675 $2910(18.14111 1 28)229 224 361)510 361 3624)14900 14496 3628)4040 3628 412 363 49 36 Here we proceed by the rule till four digits are found. We then add zero to 404 the remainder, and proceed to divide by the divisor 3628, as in the division of decimals in its shortened form. The first six decimal places in the nearest approximation to the root, [art.167,] are 141113, so that our re- sult is true for five places. So; to find the cube root of 869243.13. Pointing it becomes 869243.130. 869243130 | 95.436335591 729 140243 128375 11868130 10875664 992466000 85869 273120669 9 819362007 acsneipeeiateeenneeronme te ee eS 27320654700 (173103993000 1717776 27322372476 36 27324090288 85889 27324176177 27324262066 86 273242706 163934234856 81972528531 9725052909 819728118 273242792 $152777173 12662140 1615577 136621 24937 2459 35 2 ARITHMETIC AND ALGEBRA, _ Here we proceed as the rule directs, till five digits are found ; we then add zero to the remainder, and continue to divide the number so formed and marked & by the new imperfect divisor marked a. This division is inthe shortened method [art. 167.) To come to an end the sooner and have less cumbrous num- bers to deal with, we cut one digit off from every successive remainder, and diminish the divisor by two digits, and the’correction by three digits, at every division, instead of diminishing the divisor at every step by one digit, and the correction by one. ‘This result is accurate to the last decimal place.* 186. Similar rules may be given for finding the fourth, fifth, and other roots of numbers. The operations, however, become extremely laborious, and the result can always be found with suffi- cient accuracy by means of a table of logarithms. The rules for finding the square and eube roots of algebraical expressions are founded on the same principles as those just laid down for numbers, but as there is never any occasion to em- ploy them they are of no practical use. 187. By the rules we have laid down we can find a number consisting of as many decimal places as we please, which number is always less than the square root or cube root or other surd sought. Now, by increasing the last digit of this number by unity, we find a number which is greater than the root sought; for the process by which we obtain each digit assures us that it is the greatest digit that is not too great. We can thus always present two num- bers, one greater and the other less than the surd sought, while these two numbers may be made to differ from each other by a quantity as small as we please. For instance, we have found fart. 185], that 95.436335 is less than the cube root of 869243.13, while 95.436336 is greater than the same root, and these numbers differ only by ge. Bey of unity. If we wish to find 1000060 two numbers which differ only by 1 10000000 , between them the same root, we do it of unity, and which include * The rnle we have given for the cube root ismuch Jess laborious than the one commonly used. It was first given in a Treatise on the Numerical Solution of Lqnations, by Mr, Holdred, 57 by carrying the process one step farther, and find the numbers 95.4363355, and 95.4363356. 188. We can now explain what. is meant by the product of two or more surds. Take, for instance, »/a and 34/6. Let s be a number greater, and s’ less than ,/a ; and let ¢ be a number greater, and ¢’ a number less than 3,/0. The product ./a.%,/b means a number which is always intermediate between st and s'd, however small the differ- ence made between s and s’, and between ¢ and ?#’, or, in other words, however small the difference be made between s¢ and s! #’. 189. We may extend the proposition in art. [23], which was proved in art. [89] to be true for fractions, to the case of surds. Retaining the notation of the last article, the product Ja. a/b is a number that is always intermediate between sz¢and s’#’; and 3b. fais a number intermediate between ¢s and t's’. But sé and s‘?’ are respectively equal to fs and?’ s’, since s, Z, s’, and 7! are all fractional numbers, [89]. It follows that ,/a.*,/band?,/6. /a are always intermediate between the two numbers sé and s'#’, while the difference of these two numbers may be made as small as any number that can be named. Therefore the difference be- tween »/a .3/b and #,/b. Ja may be made less than any number that can be named, and hence, on the reasoning in art. [152], these quantities are accu- rately equal. The same reasoning may plainly be extended to the product of more surds than two. 190. The root of any number made up of factors, is the same as the pro- duct of the roots of the separate fac- tors. For example, Nab = Min ALU. For multiplying Ja. ./b by itself, the product is AL aalD Bala af OS and, as we have just seen [art. 189], this is equal to Mim alle A Oo s'Af Os orab, Therefore a. ./6 is a num- ber which, when multiplied by itself, produces a 6, and is therefore equal to Vaéb, [art. 11]. Thus 144 or J9x 16 = J9.f16=3%4, or 12. Similarly, m Jabe=™ af a AL é nara fd % 191. When ,/a and 3,/2 are surds, 58° Salta | the expression serene a qnapuy such that its product by ?,/6 is always included between the same numbers as include between them ,/a, however small the difference between these numbers be made. [See art. 10.] 192. The root of any fraction, is the root of its numerator divided by the root of its denominator. For, by art. "Ja. ("»/a)” Jb (fb) or is the number [90], the 2” power of ] sa a" Therefore fb which when multiplied by itself pro- a duces — 3° that is [art. 11] the x root a of —. Thus the square root of atl is Z . b ib 2 oa _ 193. The x” root of a” is a”. This follows from art. [24j, where it was shown that (a”)"=a™, So that in general when p is a multiple of x, Pp. "Ja? = a”, Thus, *4/3° is equal to 3% or 32, or 9; accordingly we shall find that the cube of 9, and the sixth power of 3 are both 729. 194, When, in the last article, p is not a multiple of 2, let p=nq +r. Then we shall have n ner. # a? = "G8. For since ae eS at ge by art. [24]; it follows that "Al ai tT = * Jao But, by art. [190], "Al ata” = "AJ a. al and we have just seen [art. 93], that "Al at = a'!, therefore wal on gr = 91. * J ar ae vam a Va! Aa", Mae = "NGF = 8a, and therefore ‘ SR. = al ® hiat Thus bay ao = Dt al because 9=2x4+1. 195. We can sometimes use the pro- perty proved in the last article to reduce two different surds to two expressions that have the same irrational part. Thus »/ 8 and 4/18 are severally equal ARITHMETIC AND ALGEBRA. to f/2x 4 and 72x 9, or to 22 and 3/2. It follows that ,/8 + ./18 =(2+3)/2=5 /2; and this again is equivalent to ,/ 25. 4/2 or »/ 50. 196. Them power of *,/ ais” a”. For this m power is the product "A Ge Al fat Al BoC. repeated m times, and by [190] this pro- duct is the same as “J a.a.a [m times ], oras",/ a”. Thus the fifth power of J 3 is Af 3°, or by [art. 185] 32. 4/3. 197. Them“ root of ”,/ ais” / a. For let" / a = b, then since the power of one quantity is equal to the same power of an equal quantity, a= 6". Again, the n* roots of equal quantities are equal; and therefore NS wy os = fm [art. 193]. It follows that 5 or "A/a is a quantity such that its 7 power is equal to ”,/ a, which is what we pro- posed to prove. For instance BP iA] 2S KF: 198. The product ™/a. */ @ is m fant. For, observing that 6 and ",/ 6" are the same, and then putting ™ / a for 6b, we have ae Ga tie (PR) tay Again, by [art. 196], Cf a)” =s © af of", therefore aN Ae nal Mailed or by [art. 197] hs = ie hy ae 7. (A), similarly Nil ea atm, iy CCB Therefore, multiplying these equations together, Tic SAP a= Se AL a”. ae: But by [art. 190] the second member of the last equation is equal to mn af qm, Os therefore PR) Me "Ja = ml am gn, or mn Af grr, Thus ‘ 3/3. 4/3 = 4:3 4/3443 12 4/ 37 = 12/9187. _ 199. Similarly dividing the two equa- _ tions (A) and (B) in the last article by each*other, we have ¥ ARITHMETIC AND ALGEBRA, mu al a) ch mie J a” : n af a mn Af a” or by [art. 192] m AJ a mn q” Ja Ne ey a = —- =x Sal 3s 8A 3 32 J m * a Thus These results seem to be extremely complicated. We shall find. however, when we come to explain fractional and negative exponents, that they are capa- ble of being expressed very generally and simply. 200. By [30], the square of — ais @, the fourth power of — a is at, its sixth power is a6, and so on. Therefore aand — a are square roots of a?, fourth roots of a4, sixth roots of a®, and so on. Every number, in like manner, has two square roots, two fourth roots, two sixth roots, and in general two of every root indicated by an even num- ber, and these two roots differ only in their sign. In writing such roots, when there is nothing to determine us which of them is to be taken, we ought to write them both; and this is done by prefixing both the positive and negative sign, or the dowdle sign, to the positive root. Thus /9 is +3; the fourth root of ais + 4+,f a. is read plus or minus. . 201. A positive quantity multiplied by itself, and a negative quantity mul- tiplied by itself, both give a positive product ; there is therefore no quantity positive or negative, which when mul- tiplied by itself gives a negative pro- duct. It follows that a negative quan- tity cannot have for square root a ‘quantity either positive or negative. The square root of a negative quantity such as 4 —a is therefore called an imaginary, or an impossible quantity. Whenever such a quantity appears in a result, it is certain that there is some- thing contradictory or absurd in the conditions of the question. In the same manner, since every power indi- cated by an even number of a negative quantity.is positive, no negative quan- tity can have any root indicated by an “eyen number that is not impossible. The quantity 2/_ @ is therefore al- ways imaginary. Quantities not 1ma- ginary, such as 4, Ja, &c. are some- times called possible or real quantities. 59 202. Since V—a = Va.(—1), we have by art. [190] V—a= fax V—1, In the same way, any impossible square root may be written in the form m7 —1 where m-is possible. —1 means an expression which when multiplied by itself produces —1. Therefore . (/ —12=—-1, (V=18 = (V—15. = ein aL, (Vo 1)4=(V = 1). Vo 1 =e ttf mda af ea In like manner we have , * (m Al 12 = — m?, V1 and so on. (m V—1)3 = —m. aes (m Ni — 1)4 ra tan (m J—1)8= +m, o, (m J —1)6 = — m6, (m V1) =—m'. Ate and soon. So that all the powers of an impossible square root can be writ- ten in the form 2 4/—1, where 2m is possible. In the same way it may be shown that 4/4 —a, or any other impossible fourth root and all its powers can be written in the form aN al. a It may be observed, that 44/ —.1 is es- sentially different from A —1, since if we raise both these expressions to the fourth power the results are dif- ferent, namely, — 1 and ++ 1. 203. If we cube oe eas 2 a, the result in either case will be a3. These two expressions, then, as well as a, are cube roots of a’. We have already seen, that a2 has two square roots, a and — a. It will be shown afterwards, that every quantity has four fourth roots, five fifth roots, and so on. But of all the roots of any quantity mdicated by an 60 ARITHMETIC AND ALGEBRA. odd number, one only is real, and all the others imaginary. So of all the roots indicated by an even number, all are imaginary but two, and these two differ only in their sign. . Of Quadratic Equations. 204. Let it be proposed to find two numbers such that their difference is 8, and their product 48. Calling the one number a, the other will be 8 + a, and taking their product we have “x (8+ 7) = 48, or 2 +8 & = -A8, It will be found, that the circtimstance of 2? occurring in this equation prevents us from solving it by the rules laid down for the solution of simple equa- tions [111.] To each member of it let us add 16. It then becomes w+S8ar+16 =48 + 16 = 64, Now observe, that the first member is here the square of «+4, as may be proved by multiplication. Therefore (w+ 4)? = 64 = 82. When two quantities are equal, their square roots are also equal, and there- fore e+ 44> +8, fart, 201], whence x= 8 —4, or4, if the positive sign be taken; and x= — 8 —4, or — 12, if the negative sign be taken. Either of these numbers will satisfy the ques- tion. If 4 be taken the other number is 4+ 8 or 12; we find that the dif- ference of 12 and 4 is 8, and their pro- duct is 48, Similarly, if — 12 be taken, the other number is — 12 + 8 or —4; the difference of —4 and — 12 is 8, and their product also is 48. In taking the square roots in this example, the double sign should have been affixed to each member, and we should have had t(ie7+4)=+8. This expression furnishes four equa- tions; but of these, the two produced by giving the first member the negative sign, namely, —(e@+4)=8, and —-@w+4=—8, are equivalent to the two produced by giving it the positive sign. The double sign therefore need be prefixed to one member only, 205. Every equation which, like the one just solved, can be put into the form dy a G = 0; [see art. 109], where p and q may be any quantities positive or negative, is called a quadratic equation. Quadratic is derived from a Latin word meaning a square, and the equation is so called because the square of x. occurs in it. All equations in which z is found in its first power and its square only, can be put into this form. Thus the equation ax+ba+c=a2+d, by removing a? to its first member and c to its second, becomes (6—1)a2+ar=d~—e, and, dividing both members by 6 — 1, a fear uz + See bed Phy? or a aoe ax? C — ———_—_— — : Ey appeey | 05 and this last is of the form required. _ The equation a? + q = 0 isa quadra- tic equation ; for, algebraically speaking, it is of the form given above when p, the coefficient of x, is made zero, 206. To solve generally the quadra- tic equation . a+ prt q= 0, remove q into the second member, and we have x? + xe = —q, 2 Now a+ px+ i Is in every case the square of 2 + £. To each mem- i 2 ber of the equation then add (2 ) , or P . ate and it becomes at + px + (2) = a qs or (e+ Bae As, in art. [204], the square roots of these equal quantities are equal; the square root of (w + ) is a (x ee ARITHMETIC AND ALGEBRA, 61 : 2 2 and that of & —gq ist Cy ys therefore | whence pn Egan ay, ae: When the equation is of the form w+q=0, or a= —q, taking the square root of each member as before we have gat —q....(B)- These expressions (A) and (B) are quite general, and contain in them- selves the rules for solving quadratic equations, whatever the values of p and gq may be. For instance, the equation in art. [204] is the same as g2+ 8x —48=0, where p = 8 and q = —48. Therefore 2 £ = 4 and 7 = 16, so that the expres- sion (A) becomes e=—4+ Vi6— (—48), = —-4+ J64=—448, as before. Observe, that when zero is put for p in the expression (A), that expres- sion is reduced to # = = Vi q which is the expression (B). 207. The values of the unknown quantity which solve, or satisfy, a qua- dratic equation are called its roois, because they are found by the extrac- tion of a root. The expressions just found show us that every quadratic equation has got two of these roots : V12. mnt a/ Ee ae 4 qd» Bo aff og isis 12 4 qs in the case of the form (A); and + AV —q, and — J — q in the caseof the form (B). We may show that it can have but two. Take the equation a+pxtq= 0, and let a be one root of it; then a+ pa+ q=0- or and Again, let a’ be another root, then as be.ore, a* > pa + p= Ol Subtracting the second of these equa- tions from the first we have @—a*+p(a—a’)=0, or a@—a%®= —p(a—d). Divide both members of this equation by a — a’, and it becomes at+a@d=-—p, or /=—(p+a). It is quite plain that there is only one quantity equal to —(p+a), and there- fore the equation has only one root besides a. Learners sometimes find it difficult to understand how the same quantity can have two different values in the same equation. This will be fully explained afterwards. It is enough to say at pre- sent, that either of these roots is always found to satisfy the equation, and therefore if our expression did not fur- nish us with both of them, it would not solve the equation completely. 208. The sum of the roots of the equation 7 + px + q=0is gee which reduces itself to —p. Their product is {5-0 aap Comparing this with the second exam- ple in art. [28] we find it equivalent to ed eta) (2) which reduces itself to qg. _ Thus the sum of 4 and — 12, the tw roots of the equation in art. [195], 1s — 8, and their product is —48. We have seen [art. 197] that the values of p and q in that equation are 8 and — 48 respectively. We can thus form a quadratic equa- or 62 tion whose roots shall be any given numbers, 2 and 3 for instance. We must make —p=2 + 3, ip = and q=2x 8, or 6. The equation then is M—5e¢+6 = 0. 209. The rule for solving quadratic equations is as follows: Clear the equation of fractions if there be any [110]; remove all the terms containing the unknown quantity into one mem- ber, and all those that do not contain tt anto another; collect into one the co- efficients of the square of the unknown quantity if more than one, and also the coefficients of the simple power of the unknown quantity ; divide every term in both members by the coefficient of the square of the unknown quantity ; to each member of the equation as tt now stands add the square of half the coefficient of the simple power of the unknown quantity ; extract the square root of each member, and write the results as equal to each other, prefixing the double sign to the one which does not contain the unknown quantity ; the quadratic equation is then reduced to a simple one, and may be solved accord- ingly. 210. The following are examples of questions producing quadratic equa- tions. The sum of two numbers is 6, and : ‘ ye: the sum of their reciprocals is rr what are the numbers? Call the one of them x2; then, as in art.[115], the other is 6 — x, and taking the sum of their reciprocals [96] we have by the ques- tion Le 4] 1 3} x 6—2 An Multiplying both members of this equa- tion by x (6 — #) it becomes g ie +x >=—(64%— 72), 6-a“- ra ) or 3 — (6% — 7) =6 ri ( ) 7 4 and multiplying both members by — se a—62XL= — 8B, Gh Fil To each member add (=) , or 9,and we have ARITHMETIC AND ALGEBRA. a —6e7+9= o— 8, or (v— 3 =1, whence - Bo 'S eA 1: and this gives us C= 3 i, or 2 =4, or 2. If 4 be taken for the one number, the other is 6 — 4, or 2; and 2 and 4 are the numbers that satisfy the question 1 3 ' 1 since —-+ —= —. 2 -: 4 4 It is observable, that in this case the two roots of the equation 2 and 4 are the two numbers sought. This is ne- cessarily the case ; for whichever of the numbers is called x, the other is 6 — 2, and there is no way of distinguishing that # is to stand for one of the num- bers rather than the other. Every rea- son, then, that exists to make one of the numbers satisfy the equation applies equally to the other. It is therefore impossible that the equation can be satisfied by one of the numbers and not by the other. 211. To find two numbers such that their sum is 10, and the sum of their squares 58. Call the one number z, then the other is 10 ~#. Bythe ques- tion “+ (10 — xv = 58, or 40 =~ 2027+ 100 = 58. Carrying 100 to the other member of the equation and dividing by 2 this be- comes x —10x2= — 2), Add 25 to each member and we have VE S10 er 95 — 21, or tee (© — 52 =4; whence ino tes Ti an, | which gives’ 7 =8, or 7.. We find 3% -+ 72? =9+49=58. For the same reason as in the last example, the two roots are the two numbers sought. Suppose that it had been required to find two numbers such that while theiy sum is 10, the sum of their squares should be 20. As before, we should have | 2a2 — 20x + 100 = 20, and w— i102 = — 40, oe ~ Seven ARITHMETIC AND ALGEBRA. Adding 25 to each member, this be- comes ’ vw —10%+25 = — 15, which gives us -w=5+N — 15, or e=5— NA — 15. Here the values of x are impossible fart. 201], which shows that there are no numbers that can satisfy the con- ditions given, but that the question contains something absurd and con- . tradictory. This is plainly the case; _ for we cannot divide 10 into two parts such that the sum of their squares shall be less than 50. The imaginary values of w however satisfy the question, and, as before, the two roots of the equation are the two expressions sought. Returning to [206] the roots of a quadratic equation will be impossible 2 whenever © — q is a negative quantity. 2 Now F is in its nature positive, since every square number is always posi- tive. Therefore the roots are impos- sible only when q is positive, and greater p than a When q is a negative quan- tity, such as — m, the expression under LP the radical sign becomes ae (—m), or 2 r +m, a quantity essentially positive, and therefore when gq is negative the roots are always possible. If one root be impossible the other must be impos- sible also, for the two roots always con- sist of the same terms connected by different signs. 212. A company of persons spend 31. 10s. at a tavern. Four of them go away without paying, in consequence of which each of the others has to pay 2s. more than his share. How many per- sons were there in the company, and what was the proper share of each? Call the number of persons #. They spend 70 shillings, so that the proper 4 AY ALS ae share of each is “ shillings. But there are only x — 4 who pay, so that every 0 ; one of these pays Price shillings. Now 63 each of these pays 2s. more than_ his proper share, therefore 70 Multiply each member by #, (a — 4) and this becomes 70% =702% — 280 +222 8a, or, when reduced to the general form, | x2 —A4a= 140. Adding 4 to each member we have “2 —4e4+4= 144, which gives ies Zea et Ag: whence x = 14, or — 10. The positive value of «, 14, satisfies the question; for when the number in com- pany is 14 the proper share of each is 70 | aie 5 shillings, while what each of those who remain actually pays is rir or 7 shillings, which is 2 shillings more. With respect to the negative value, — 10, it also. satisfies the equation, since : 70 79 tet Ne ty (LO Seo —10—4 — 10 Rt As we have stated the question, the 70 70 : sums and — are to be paid b ae 4 x L y the company, when they are positive numbers. Therefore when they become negative numbers, that is, when # be- comes negative, they are sums to be received by the company. But when we introduced w into the equation, we introduced it quite generally, as any quantity either positive or negative that would satisfy the equation ; and there- fore the equation, as we stated it, neces- sarily applied as much to the case in which the company were to receive, as to that in which they were to pay the money. When they are to receive the money the question must be altered to this: A company of persons are entitled to have 70 shillings distributed among them, their number is increased by four, in consequence of which the share of each is diminished by two shillings ; how many persons were there at first ? This question is, algebraically speaking, the same as the former, with the signs of the different numbers changed ; the answer to it is 10, the negative answer to the former question with its sign changed.* Not only, then, do the two 64 roots satisfy the original equation nu- merically, but they both satisfy the con- ditions of the question in the way in which these conditions must be trans- lated into the language of algebra. 213. When there are two or more quadratic equations involving two or more unknown quantities, there is no general rule that can be laid down for their solution, and we must treat them differently, according to the circum- stances of each case. For instance, let it be required to find two numbers ahead whose product 1s i and the sum of ‘ 17- their squares ae Call the one a, and the other y. Then 1 Y= Serres (A), 1 BO OP es (BD. Multiply (A) by 2 and it becomes 2 2x Y= 5% Adding the members of this last equa- tion to those of (B) we obtain Spa. Ng oes, il alld Pat BOTA Wa Fiaigg 89-7) 35? and, in like manner, subtracting them, PR ORE we 28 y TPH ae oe oF Now, as we have seen, e+ 2eytyr=(a+y)%, and similarly v—2ry—yr=(w—y)* Therefore + yy = 4 y)2 = — ) z om as Sand amg and extracting the square roots of both members of each of these last equations eby= ok | eo osleox Do fs mee Adding these equations, and attending to all the signs, we have 4 1 = -, or +-, 22 ie see and 36 1 c= an <5 or Bye vo ARITHMETIC AND ALGEBRA, Similarly by subtracting, 1 2 y= t-,ort—. y= bo, orcks 2 1 The numbers sought then are 5 and ae 2 I : ‘ or — 3 and ew either of these pairs satisfies the question. The values of x and y are the same, because the equations (A) and (B) are symmetrical with respect to v and y. [See art. 112.] 214, Find two numbers such that their sum, their product, and the dif- ference of their squares, shall be all equal. As before, let the one be xv and the other y. Their sum isa + y, their product wy, and the difference of their squares a® — y%, Therefore by the question cty= a —y?*, Lry=LY. Dividing each member of the first of — these equations by @ + y we find l=av-y, whence Yi — 1. ' Substitute this value for y in the se- cond equation, and it becomes Ce —1S 2(e— 1), or e-3xe7=— 1, and the roots of this quadratic, deter- mined by the rule, are 3 5 is C= soe and @ = 2— ME. The corresponding values of y, found by putting these values for w in the equation y=xr-—-1 1+ 5 2 are Yy — Accordingly, either of the pairs of num- eae dnd-vae2 or ; ae 3 1-— i ey peel bers and fireghs will be found to satisfy the conditions given. We may observe, that the two equa tions which we have just solved are not symmetrical ; because, though they con- tain the same powers of x ani y, the sign of y? 1s different from that of 2%. Accordingly, the values of x and y are not the same, as they always are when the equations are symmetrical. ARITHMETIC AND ALGEBRA. 215. Again let us have the two gene- ral equations ‘ art by? cxy=d ear byt gory dl. (A) and let it be required to find the values of x andy. Suppose that Y= zu) that is, instead of y write 2 #, where z is an unknown quantity. Then the equations become ax + be2a®+czatad a'n® + b' 2%42 + oz x2 = dl, Divide the members of the first equa- tion by those of the other, as in art. [139], and we have Ge? be ater oan oF nae —————_——_ = — >» alae + b! 2242+ 242 dd! and striking the common factor 22 out of the numerator and denominator of the left-hand equation, Ga 623; ez "d a Oat Pa qth Getting rid of the fractions in this last equation, it assumes the general form of a quadratic equation [art. 205], and may be solved accordingly. When we have found the value of z as we substitute it in equation (A), 2 will be the only unknown quantity in this equation, and it may therefore be solved. When we know # and 2 we find y from the equation Y=HLeHr, If we observe the two original equa- tions, we shall see that the sum of the indices of # and y is the same in every term where they occur. It was for this reason that the substitution y = za was had recourse to, which would other- wise have been of no use, as the suc- cess of the method depends upon the same power of x appearing in the seve- ral terms, so that a may disappear on the reduction of the fraction. In all cases where the same condition is ful- filled whatever be the number of equa- tions and unknown quantities, we ma by a similar method of substitution re- duce the number of both by one. Thus take the three equations involving 2, y, and 2, : axz® -vy7e= A, cy=+dzr=B; ezxt+ fay =C, 65 Let Y = UL: and 2= 02. Substituting aa + buv.a2?= A, cu2,a2+dv.a2?=B, GU" .2? > Ua? = G, Dividing the second and third by the first, and striking out 22, which will be common to the numerators and deno- minators of both fractions, we have cue+dv B atbuv ' of 8 re hy Bi or 34 910, or 31024, or 1024”. Secondly ; To divide any power or root, direct or inverse, of a quantity, by any other power or root, direct or inverse, of the same quantity, subtract the exponent of the divisor from that of the dividend, the result is the exponent of the quotient. Thus the quotient of 4/3, divided by 3, where the difference of the expo- 2 . 1 nents is — — — 3 Thirdly ; To find any power, or root, of uny power, or root of any quantity, multiply the exponent of the quantity by the number which is the exponent of the power or root required. Thus the 4 cube of a” is a’, ora®, The square root -4-1 -2 2 5 4.0%, 0us ’ Cal 1 - t of —, or of a’, iS @ a 231. These rules contain every thing that it is of much importance to attend to in the arithmetic of surds, but they apply to those surds only in which the same quantity is affected with different exponents. When different surds are to be added to or subtracted from each other, or when the product or quotient of surds in which the quantities affected with fractional indices are not the same is to be taken, we cannot, in most instances, do more than indicate these operations by the proper signs. Sometimes, how- ever, the results of these operations admit of being simplified a good deal. If we take any whole number N, and reduce it into its prime factors [art. 62], we get a result either of the form a”, where a isa prime number and m either unity, or some other whole number, or of the form a™ a”, where a and a’ are dif- ferent primes, and m and! whole num- bers, or of the form a”. a’. a”, or some 69 similar form. Now, if we take the n‘* root of N, we shall always find it to be a surd, unless ‘all the exponents m, m’, m', &c. are multiples of x, Thus 360° is a surd, because 360 = 23.32.53; -and3 and 1, two of the exponents, are not multiples of 2. This may be proved as follows: let us suppose that m, m’, m', &c. are not 1 multiples of’, and still that N” is a whole number. Let us reduce this whole number into its prime factors, so that we find 1 ; Wi OF Oe OU Raising both members of this equation to the 2 power, we have N35 bt b te De. But by our supposition, N = Co qi™, gm. ee 3 where m, m’, m’, &c. not being all multiples of m, are some of them different from xp, np’, np", &c. which are all multiples of 2. We, therefore, have reduced N into two different sets of prime factors, which [art. 61] cannot be done. It follows that t N” cannot be a whole number. When m, m’, m’, &c. are not multi- ples of 2, we have, by [art. 190], 1 m m mit N= a" a, a, 0, and if m=qn+r,m =qn+7, &e. we haye, by [art. 194], ¢ 1 ? 7 . N” at atts alte al't’ta. . 5 eee ys yt = at.at.al’x..a.a™.a™,, Thus 2 i a 1 360° = 23.31.27 .5°, or 6.10. 232. In this way, when we have a number affected with a fractional expo- nent, we can, at once, find whether it be a surd, and if so we can reduce it into its simplest form. When surds are to be added or subtracted, if, when they are reduced into their simplest forms, the surd parts are the same in each, their sum or difference can be reduced to one surd, as in art. [195]. Again, when surds are to be multiplied or di- vided by each other, by reducing them into their simplest forms, we Can per- 70 ceive at once whether the résult will admit of any material simplification. Thus, to take the product of 216° by 108*.. We find 108 = 22.3%, and there- fore 108" = 2.3%, Again, 216 = 29, 33, and therefore 216° = 9? : 3, The pro- duct sought is therefore 2 3 gts Or 2.3%) or2x3 x 3%, or 6 Xx 3%, Si- milarly to divide 48? by 727, we find 48 = 24.3, and therefore 48? = 92.3": so 72 = 2.3% and 72 = 2*.3', The i : oP ae 22 quotient; therefore, is WGconkGl = y* 37 4 -2— 3: ne 3 2 Marea | sees . 2% 233. Since [art. 224] if the exponent of any quantity be a fraction we may substitute for it any equivalent fraction, and since there is a decimal fraction equivalent to every vulgar fraction, we may always use decimal fractions to express the fractional exponents of re quantities. Thus a” is the same as a’° t 1 2 So a! is the same as a’, Orin. < * 5 85 is the same as a”, and so on. When we find for the exponent of a quantity a repeating decimal, or one consisting of many places, we may cut it short, as in art. [166], without intro- ducing any sensible or material error into our result. Thus, if instead of sel C ate we write a we make the exponent too great by - 00000029 nearly. a nas ; oe Instead of being equal to oe then, our 34. 00000029 expression is equal to a nearly, 5 or to a7 x a+, Now a’ = 1 [art. 216], and therefore q * 929 differs from unity by a very small quantity, so small that it may be neglected,* and if it be, 5 5 5 we have a’. q ‘00009 — gq? x } oy gq’. * Ata future part of this treatise we shall inquire into the actual magnitude of such quantities as that in the text, and shall prove rigorously that they may be neglected, : _ ARITHMETIC AND ALGEBRA. Of Logarithmic Arithmetic. 234. When we have the equation ™m™ — m7 7 a” = N, we have seen that pe is called the exponent of a. When we speak of < however with reference to N, as in- dicating the power of a that is equal to N, it is called the logarithm of N. The exponent of a then, and the logarithm of N, mean the same quantity, and we use the one or the other of these names for it, according as we speak of it with reference to a or to N. To express what the quantity a is, with respect to which = is the logarithm of N, we say that ~ is the logarithm of N to the base a. Similarly if a’” = N, t R ~ would be-the logarithm of N to the 1 base a’. Thus, since 9° = 3, 1 or .5 is the logarithm of 3 to the base 9. So since 102 = 100, 2 is the logarithm of 100 to the base 10. Most of the purposes of logarithms are best served when the number 10 is the base. Logarithms to this base are therefore the most frequently used, and are called common logarithms; they are always expressed by decimals. When we write log. 5 = 0.69897, we mean that the common logarithm of 5 is 0.69897; that is, that 5 = iH tbs seaa fy or 5 10's. 5, 235. The word logarithms is com- posed of two Greek words, that respec« tively mean ratio and number, so that it literally signifies the ratios of num- bers. We have seen [art. 129] that a? is said to have to unity the duplicate or double ratio of a to unity, a® the tripli- cate cr threefold ratio, andso on. The numbers 2, 3, &c. (which are the loga- rithms of a®, a’, &c. to the base a) thus actually express the ratios of a2, a’, &c. to unity, as compared with the ratio. of a, to unity, and are therefore, with propriety, called the logarithms of these quantities. 236. Our present purpose is to ex- plain how logarithms are used to abridge the operations of arithmetic, and we shall reserve till afterwards the ARITHMETIC AND ALGEBRA, explanation of the way in which the logarithm of any number to any base may be found. We shall therefore sup- pose that we have got ready formed a table of logarithms of the usual extent ; that is, a table containing the logarithms of all whole numbers less than 100,000, or that consist of less than six digits. Very few of these can be expressed in terminating decimals, but this causes little inconvenience, since a logarithm carried to six or seven decimal digits is sufficiently exact for all common pur- poses. [art. 233.] =: 237. That part of any logarithm which stands to the left of the decimal point, is called the characteristic of the logarithm. Thus in the table we find log. 75293 = 4. 8767546, in which, consequently, 4 is the charac- teristic. Since 10° = i, by art. [216], and since 10' = 10, the logarithms of 1 and 10 are respectively 0 and 1. The loga- rithm of every number between 1 and 10 is therefore between 0 and 1, and so has zero for its characteristic. Thus log. 2 = 0.3010300. So since 102 = 100, the logarithm of 100 is 2; and the logarithm of every number between 10 and 100, or which has two digits to the left of the decimal point, has unity for its characteristic. Similarly, the characteristic of the loga- rithm of every number between 100 and 1000 is 2; and generally is the charac- teristic of the logarithm of every num- ber between 10” and 10"+, or which has 2 + 1 digits to the left of the deci- mal point. of the logarithm of any number greater than unity we have, therefore, the follow- ing rule: If the number be an integer, the characteristic of tts logarithm is the number of digits of which it con- sists, diminished by unity ; uf part of the number be a decimal, the character- istic is the number of digits to the left of the decimal point diminished by unity. _ 238. Let us suppose that 10" = N, = 10N, ua $$) Then, by art. [225], 10” m4+2 "4p 10” =100N, and generally 16” =10?.N. That is to say, that ~, which is the logarithm of N, becomes that of 10N To find the characteristic . 71 by adding unity to it, of 100N_ by add- Ing 2 to it, and so on. The logarithms of the products of a number by 10 and all the powers of 10, therefore, differ from the logarithm of the number itself only in their characteristics. Thus, since ‘log. 24 = 1.3802112, it follows that log. 240 t= 2. 3802112, log. 2400 = 3.3802112, and so on. 239. From the last two articles it follows, first, that a table of logarithms need not contain their characteristics, since the rules for finding these are so simple; and, secondly, that a table of the logarithms of all the whole numbers, from 10,000 to 100,000, is sufficient to furnish the logarithms of all the whole numbers less than 100,000. Thus the logarithm of 2598 is found by prefixing 3, the proper characteristic, to the frac- tional part of the logarithm of 25980. So, since log. 80000 = 4 . 9030900, we have log. 8 = 0. 9030900. Accordingly, the tables of logarithms most in use contain the fractional parts only of the logarithms of all the whole numbers between 10,000 and 100,000, or of all the whole numbers that consist of five digits, leaving the characteristics to be supplied. 240. If we examine several consecu- tive logarithms in any part of the table, we shall find that they consist of num- bers that are very nearly in arithmetical progression. For example, we find log. 41337 = 4. 6163390, log. 41338 = 4.6163495, log. 41339 = 4. 6163600, log. 41340 = 4.6163705; where each logarithm is formed by adding .0000105 to the preceding one. This goes on till we come to log. 41387, which is formed by adding . 0000104 only to the preceding one. Thus 49 successive logarithms, from log. 41337 up to log. 41386, are in arithmetical progression. The difference of any two successive logarithms, from log. 41387 up to log. 41777, is either .0000105 or .0000104, being sometimes the one of these numbers, and sometimes the other, From log, 41337 then to log. 41777, or for 330 numbers in succession, the dif- 72 ferences of the logarithms are either 0000105 or .0000104. This regularity in the table enables us to use it for finding the logarithms of all whole numbers between 100,000 and 1,000,000, consisting of six digits. For example, if we add unity to the charac- teristics of log. 41337 and log. 41338 we find log. 413370 = 5. 6163390, and log. 413380 = 5. 6163495, where, as before, the difference’ is .0000105. Now, according to the law which we have just observed, the loga- rithms of the numbers between 413370 and 413380 must be in arithmetical progression, and must, consequently, consist of nine arithmetical means be- tween 5.6163390 and 5.6163495: and by art. [143] the common difference of the progression which these means form will be a, or .00000105, since .0000105 is the difference of the first and last terms. Accordingly, if we add this quantity to log. 413370, we ob- tain log. 413371; if we add twice this quantity we obtain log. 413372, and so on. Thus, to find log. 413375, we have log. 413375 =5 .6163390 +5 x 00000105 = 5.6163443, cutting down the result to seven decimal digits. , In the example Just given, if D be the difference between the two logarithms, the number to be added to the least of them to produce the logarithm required is re Similarly in any other case where m is the units’ digit of the number of six digits, ies is the number to be added to’ the logarithm of the first five digits. 241. In the same way we may use the table to find the logarithms of numbers consisting of seven digits. If, as before, D be used to denote the difference be- tween the logarithm of the first five digits of such a number and the loga- rithm next above it in the table, and if m be the number consisting of the last two digits of the number proposed ; then, just as before, the number to be added to the logarithm of the first five “number are 27, and ARITHMETIC AND ALGEBRA, digits is ae Thus, ifthe number pro- posed be 8561427, we find log. 85614 = 4. 9325448, log. 85615 = 4. 9325499. The difference of these logarithms is 0000051. The last two digits of our 27 x .0000051 100 = .000001377. Reducing this result to seven digits it becomes .0000014, and adding this last number to the first logarithm, and prefixing the proper cha- racteristic, we find. log. 8561427 = 6 . 9325462. We may observe, that in like manner we should find that the number to be added to the first logarithm to form the logarithm of 8561428 ought to be - 0000014 also; so that log. 8561428 = 4. 9325462, the same as before. It follows that the first seven decimal digits of the loga- rithms of- the numbers, 8561427 and 8561428, are the same; so that if we wish to distinguish them, we must use a table that contains a greater number of decimal digits than seven. | 242. The rule for finding the loga- rithm of a number that consists of six or seven digits is therefore this: Find in the tavle the logarithm of the first five digits ; find the difference between this logarithm and the next greater one in the table; if the number proposed con- sist of six digits, multiply the difference by the units’ digit of the number, and divide the product by 10; or if the num- ber proposed consist of seven digits, mul- tiply the difference by the last two digits of the number, and divide the product by 100; adding the quotient in either case to the logarithm first found: the sum with its proper characteristic is the loga- rithm required. _In the margin of all tables of loga- rithms, the difference of the successive logarithms in that part of the tables is set down. In some tables also there are little tables in the margin, called tables of proportional parts. These are placed under every successive difference and contain for that difference the num- ber to be added in respect of each units’ digit, so as to form the logarithms of numbers of six digits. These numbers are found as is directed by the rule given above. a ARITHMETIC AND ALGEBRA. 243. We can now find the logarithms of all whole numbers of six or seven digits or under. It is easy to deduce from them the logarithms of all num- bers greater than unity that consist of six or seven or fewer digits, of which any part is a decimal. For instance, since 873043 = 10000 x 87.3043, the logarithm of 873043 differs from that of 87 .3043 only in its characteristic, [art. 238]; also the charaeteristic of the logarithm of 87 . 3043 is unity [art. 237]. Now log. 873043 = 5. 9410356, and therefore log. 87.3043 = 1.9410356. In the same way the logarithm of any other number greater than unity, and consisting of six or seven digits, may be found ; the rule is this: Find in the table the logarithm of the number proposed considered as a whole number ; prefix the proper characteristic to the decimal part of this logarithm ; the result ts the logarithm sought. 244, With respect to the logarithms of numbers less than unity, we must observe that the logarithm of every such number is negative. For as we haye seen [art. 216] the logarithm of as 1 unity is zero, and since .1] or 7) = 10-}, the logarithm of .1 is —1. Therefore the logarithm of every num- ; ] ber between unity and To’ or between unity and .1, is between zero and — 1. ] j j e ——— 1 Sat h Again, simce .01, or ida 0-2, the logarithm of .01 is — 2, and, therefore, the logarithm of every number between 1 1 ‘ — and -——, that is, between .1 and. 01, 10 100 is between —1 and —2. Similarly, the logarithm of every number between .01 and . 001 is between — 2 and — 3; and so on. Let us now propose to find the loga- rithm of such a number as 0.7434. F : 7434 7434 This number is equal to TT Uke Uae Now we find log. 7434 = 3.8712226, and therefore 7434 = 1038712226; 73 consequently ] ()3-8712228 7434 = ——, 10* By art. [226] this last number becomes 103 saee— 4" or 1 Q—0-1287774, Since 3.8712226 — 4'= — 1287774, it follows that log. .7434 = — .1287774, a negative number. We shall soon find that if is much more convenient to express negative logarithms, such as this, in a way simi- lar to that used in art. [22]. Thus 3.8712226 —4 =.8712226 —4 +3 . 8712226 — 1 = 1,8712226. i{ So that 3.87 12226—4 — 1.8712226 ] 0 —e 1 0 9 log. .7434 = 1, 8712226, where the characteristic alone is nega- tive. Similarly to find the logarithm Of 40263573. Since . 0263573 = 263573 263573 SS FTO SP Yr ———= 10000000 107 be that of 263573, with 7 subtracted from its characteristic. Now we find log. 263573 = 5.4209010, and therefore log. . 0263573 = 2.4209010. If we always bear in mind that such , its logarithm must an expression as 2.4209010 means .4209010 —2, this ‘notation cannot cause any confusion. 245. The logarithm of every number between unity and .1 being, as we have seen, between zero and — 1, is equal to — 1 with some number less than unity added to it. Its negative characteristic is therefore 1. Similarly, since the loga- rithm of every number between .1 and .01 is between — 1 and —2, it is equal to — 2 with some number less than unity added to it; its negative cha- racteristic is therefore 2. Similarly, the negative characteristic of every number between .01 and .001 is 3, and so on. In general, The characteristic of the logarithm of a number less than unity is found by affixing the negative sign to the number which is greater by unity than the number of zeros which are between the decimal point and the first 74 significant digit of the number pro- posed. i 246. To find the logarithm of any number which consists of six, or seven, or fewer digits, and which is less than unity ; find in the table thelogarithm of the digits of which the number is com- posed, as if they formed a whole num- ber, and affix to this logarithm the proper characteristic. Thus, to find the logarithm of 0. 001865, we have log. 1865 = 3. 2706788, therefore i log. . 0001865 =4. 2706788. This rule follows from art. [244]. _ 247. All that has gone before is on the supposition that itis a positive num- | ber whose logarithm is sought. If we now endeavour to find the logarithm of a negative number we are led into a curious inquiry. In art. [200] it was shown that every root indicated by an even number ought to have the double if ey sign affixed to it, thus a and10°., are each of them equal to two quantities, of which the one is positive and the other ~negative. In like manner, the number m 1 represented by a", or (a”)*” ought to have the double sign; so that, in gene- ral, whenever the exponent of a number is a fraction whose denominator is an even number, there result two numbers the one positive and the other negative. Thus, in the common logarithms, = or.5 is the logarithm of the positive square root of 10, or of 3.162278, it is therefore the logarithm of — 3.162278. But as the subject of the logarithms of negative numbers is of no practical importance, we shall not pursue it fur- ther here.* , * The existence of the logarithms of negative num. bers was strenuously denied by many eminent ma- thematicians of the last century, and as strenuously asserted by others. Every discussion on this subject appears to depend entirely upon the definition of a logarithm made use of; an arithmetical one exclud- ing the notion of the logarithm uf a negative number, and an algebraical one admitting it. Thus if 10¢ means only that number which results from extract- ing arithmetically the square root of 10, it is evident that no negative number can arise; but if 102 repre- sents thatquantity which, when squared, is equal to 10, then it is equally evident that we shall have a negative as wellas a positivenumber. We may also observe that, by the same method of reasoning, though we cannot enter into it here, we may establish the existence of logarithms of impossible as well as pos- . sible quantities, ARITHMETIC AND ALGEBRA. 248, Having now explained how the logarithm of any number, whole or de- cimal, consisting of six, or seven, or fewer digits may be found in the table, it remains to show how the nunmtber cor- responding to any given logarithm con- taining seven decimal digits is to be found. If we find the given logarithm in the table, nothing is necessary further than to take the corresponding number, and place in it properly the decimal point. This is done by means of the character- istic of the logarithm, as is explained in arts. [237] and [245]. Thus if the logarithm be 2. 6976826, we find in the table log. 49852 = 4.6976826 ; the characteristic 2 shows that there are to be two zeros to the left; the number sought, therefore, is 0.049852. Let us suppose, however, that the given logarithm is 3 . 8447592, a number which is not in the table. We find in the table ' log. 6994.5 = 3. 8447567, and log. 6994.6 = 3. 8447629, These two logarithms contain between them the logarithm proposed and their difference is .0000062. The difference between the number proposed and the least of these logarithms is . 0000045. -Now, if we call the first of these dif- ferences D, and the second d, we have seen [art. 241] that ~ £2 ~ 100’ where z is the last two digits of the number sought. ‘Therefore n= 100% . i dD In our example we have : 0000045 nm = 100 ——___ = : — 0000062 aie We thus find 7 and 3 for the sixth and seventh digits of the number required, so that log. 6994.573 = 3. 8447592, _ The rule deduced from this example is as follows: Find in the table two successive logarithms which include be- - tween them the given logarithm ; find the difference between these two logarithms, and also the difference between the least of them and the given logarithm ; mul- . ARITHMETIC’ AND ALGEBRA. tiply the second of these differences by 100, and divide the product by the jirst of them; place the first two digits of the quotient to the right of the number corresponding to the least of the two logarithms found in the.table; the re- sult, with the decimal point, if neces- sary, properly placed as indicated by the characteristic of the given logarithm, ts the number sought... ~ When the characteristic of the given logarithm requires a greater number of digits to the -left of the decimal point than there are in the number found as the rule directs, we must make up the deficiency by adding a sufficient num- ber of zeros to the right. Thus, sup- pose that the given logarithm were 10. 7543756, we find log. 5.680357 = . 7543756, Now the characteristic 10 shows that the number sought has 11 digits; it therefore is 56803570000. —_. 249, By means of a table of loga- rithms the operations of arithmetic are very much abridged. Multiplication and division are performed by addition and subtraction, the powers and ,roots of numbers are found by multiplication and division.* This use of the tables is what we are now to explain. By the first rule in art. [230] the ex- ponent of the product of two or more factors is found by taking the sum of the exponents of the factors. In other words, the logarithm of the product of any numbers is the sum of the loga- rithms of the numbers. Thus, by our definition [art. 224] we have f ; b a2 1018: ie: and ; b! = 10! wv Therefore 6. = 1 Olos- 4 + los. Bb. A so that, again, by our definition, log. 6 b! = log. 6 + log. 8’. This gives us the rule for taking the product of numbers by means of a table of logarithms: Ind the loga- rithms of the several factors taken po- sitively, and add them together, attend- ing to the signs of their characteristics ; find in the table the number correspond- ing to this sum, and affix to wt the negative or the positive sign, according as an odd or an even number of the factors are negative; the result is the * See the Preliminary Treatise, p. 9. 75 product sought. Thus to find the pro- duct of 17.934, —0.077692, and 0.3257 we have log. 17.934 Il 1. 2536772, log. .077692 = 2. 8903763, log. .8957. = 1.5973661, log. | .5513401 = 1.7414196. In adding these three logarithms we con- sider 2.8903763 as if it were written . 8903763 —2. fart. 243]. The >pro- duct must be negative, since one of the factors is negative; accordingly — .5513401 is the product sought, for it is the number corresponding to the sum of the logarithms taken nega- tively. 250. By the second rule in art. [230], the exponent of the quotient of two powers of the same quantity is found by subtracting the exponent of the divisor from that of the dividend. The loga- rithm of a quotient is therefore the dif- ference of the logarithms of the dividend and divisor. For, since 8 i ap and b! s 10198 e. we have % 7 = ] (log: b=log. b” : that is log. wy It follows that to divide one number by another we have the following rule: Find the logarithms of the two numbers taken positively, and subtract that of the divisor from that of the dividend ; the number in the table corresponding to the difference, with the proper sign affixed to tt, 1s the quotient sought. Thus to divide — 0.077698 by — 0.13976 we have = log.b — log. 0’. log. .077698 = 2.8904098, log. .13976 =1.1453829, log. . 5559384 = 1. 7450269. Here we subtract (. 1453829 — 1) from (. 8904098 — 2), and, we find for their difference (. 7450269 — 1), that is 1.7450269. This is the logarithm of 5559384, which is the quotient sought. The sign is positive, because both dividend ‘and divisor were nega- tive [art. 38]. 76 The logarithm of a vulgar fraction is found by subtracting the logarithm of its denominator from that of its nume- rator. We can find the decimal cor- responding to this logarithm, and, where the numbers are large, this is the best way of reducing a vulgar fraction into a decimal. We may here remark, that since log. = = log.a — log. 6, we have, arranging the ‘terms differ- ently, log. > = — (log. b — log. a) =-— log. ome | Tie 251. By combining the last two rules we can perform at once an operation, that, by the common rules of arithmetic, would require both multiplication and division, as, for instance, the rule of three. Thus, when we have the pro- portion OED od iD y which gives [art. 133] the equation fll oh De a *] by [art. 250] we find log. b' = log. (a b) — log. a, or log. b' = log. a’ + log. 6 — log. a.’ For’example, let it be required to find a number which shall have to 93 .7624 the same proportion as 272 . 396 has to 357.698, we have log. 272.396 = 2.4352007 add log. 93.7624 = 1.9720287 4,4072294 sub. log. 357 F698 C= 2 45535 Mop log. 71.40241= 1. 8537129 The answer is 71 .40241. 252. By the third rule in art. [230], any given power of a quantity is found by multiplying its exponent by the num- ber indicating the given power. In other words, the logarithm of any given power of a quantity is found by mul- tiplying the logarithm of the simple quantity by the number indicating the power in question. Thus, since b s 1 gles: e it follows that ARITHMETIC AND ALGEBRA. Be 4. | 0)” (log.5) . so that log. (6") = 2 log. b. To raise a number to any given power, then, the rule is this: Mind the loga- rithm of the given number, and multiply it by the number indicating the given power ; the number corresponding to the product, with its proper sign, ts the power sought. Thus, to find the twen- tieth power of . 996, we have log. "996 = 1. 9982593 20 ——= log. . 9229666 = 1. 9651860. We multiply (.9982593 — 1) by 20, we find for product 1. 9651860, and this is the logarithm of .9229666, which is the number sought. , So to find the thirtieth power of 2 ;- log. 2. = .3010300 30 log. 1073741000 = 9. 0309000, the result is 1073741000, which of course is only an approach to the truth, and is not correct beyond the first six digits. 253. By the third rule in art. [230], the root of any quantity is found by di- viding its exponent by the number in- dicating the root in question; that is to say, the logarithm of any root of a quan- tity is found by dividing the simple quantity by the number indicating the root. As in the last article, we should find 1 « log. (>) cee . 2 Therefore, to find any root of a given number, we have this rule: Divide the logarithm of the given number by the number indicating the given root; the number corresponding to the quotient is the root sought. Thus to find the cube root of 10, we have 3)1 . 0000000 = log. 10 , -3333333 = log. 2.15442 ; so that 2.15442 is the number sought. When the characteristic of the loga- rithm is negative, and is not a multiple of the number indicating the given root, there is a slight change to be made in the logarithm before this division is per- formed. Let it be required, for instance, to find the square root of 0.006543. We find log. .006543 = 3,8157769. Before dividing this logarithm by 2, we ARITHMETIC AND ALGEBRA, must observe that it is equivalent to .8157769 —3, that is, to 1.8157769— 4. The half of this is .9078885 — 2, or 2.9078885, and we find log. . 0808888 = 2 .9078885. So that + . 0808888 is the root sought. If we had divided the original logarithm by 2, without this preparation, we should have had our result disturbed by a ne- gative decimal digit proceeding from the — 3. In like manner, to find the value of 5 .2°, we must multiply the logarithm of .2 by 5, and divide the product by 6; - : log. .2 = 1. 3010300 5 4.5051500. This last logarithm is the same as .5051500 — 4, or as 2.5051500 — 6, the sixth part of which is .4175250—1, or 1.4175250. Now log. . 261532 = 1.4175250. So that -L. 261532 is the number sought. 254, An equation, such as a’ = 0, where the unknown quantity is an ex- ponent, is called an exponential equa- tion. Itis solved at once by means of a table of logarithms ; for since the loga- rithms of equal quantities are equal, we have log. (a”) = log. b; but [art. 242] log. (a*) = x log. a; therefore x log. a = log. 6; whence _ log. 2 ~ log. a Thus, to solve the equation 97 = 1976, or to find the number indicating that power of 2 that is equal to 1976, we have _ log. 1976, ea ae eae log. 2 or 3.2957869, , * = —3010300’ that is, x = 10.9483669. | We may observe that, since Ch ag = 1976, the number 10.9483669 is the 17 logarithm of 1976 to the base 2. Mul- tiplying then the logarithm of the num- 1 bem by log. 2 base 10) the result is the logarithm of the number to base 2. This we shall see is only a particular case of a gene- ralrule. We may, in like manner, by means of a table of common logarithms, find the logarithm of any number to any base whatever. This is done by solving, as above, the equation a= b; where a is the given base, and 6 the number. Tt will be observed that some of the rules which we have given for the use of logarithmic tables are deduced merely by‘ observing the way in which the logarithms succeed each other. This is not a strict method of proof. A more regular demonstration of the accuracy of these rules will be given in a subse- quent chapter, in which we shall treat further of exponential quantities, such as a®, and show how a table of loga- rithms may be formed. (log. 2 being calculated to “ Of Permutations and Combinations. 255. The combinations of any num- ber of things are the different parcels that can be taken, each consisting of a certain number-of those things, without regard to the order in which they stand in the parcels. Thus, ab, ac, ad, be, bd, cd, are different combinations of the four letters a, b, c, and d, taken two at a time. Similarly, abc, abd, acd, bed, are different combinations of the same letters, taken three at a time, since every- two of these parcels consist of different letters. But the parcels cba, dba, dca, dcb, are respectively the same combinations as the last, for they consist respectively of the same letters, though these letters are differently arranged. 256. The permutations, again, of any number of things, are the different ways in which those things may be presented by varying the order in which they stand; or the different ways in which the different combinations, each con- sisting of a certain number of those things, may be presented by varying the order in which the things stand in each 78 combination. Thus, - abe, ach, bac, bea, cab, cba, are permutations of the three letters a, 5, and ce, taken all at once. Similarly, _ ab, ba, ac, ca, be, cb, are different permutations of the same letters, taken two at a time, formed by varying the order of the letters in each of the combinations ab, ac, be, 257. Let us propose to find how many permutations can be made of.any num- ber of different things, taken two at a time. Let the things, for instance, be the six letters a, 6,c..f. It is plain that if we write a, followed by each of the other five letters, b followed by each of the other five letters, and so on, we shall have formed the number of per- mutations required, and no more ; since every two of the permutations so formed are different. Now, when we have done this, we have five permutations be- ginning with a, five with 8, five with c, and so on; that is, five with each of the six letters. The whole number, there- fore, is 6 X 5, or 30. ‘In exactly the same way we may show that the number of permutations of five things, taken two at a time, is 5 x 4, or 20; and that the number of permu- tations of m things, taken two at a time, is m.(m—1). Again; let us propose to find the number of permutations of any number -of things, taken three at a time. Taking the six letters as before, if we write a, followed by all the permutations of the other five letters, taken two at a time, then do the same for J, then the same for c, and so on, it is plain, as before, that we shall have formed all the per- mutations required. Now, considering the class of these permutations that be- gins with a, we have a followed by all the permutations of five letters taken two at a time, that is, as we have seen, by 5 x 4 permutations. The number of permutations in the class beginning with 6 is in like manner 5 x 4, and so on. There are therefore six classes, each consisting of 5 x 4 permutations, and therefore the whole number of permu- tations is 6.5.4, or 120. Similarly, the number of permuta- tions of five things, taken three at a time, is 5.4.3, or 60; and of m things ARITHMETIC AND ALGEBRA, taken three at a time, is m.(m—1).(m — 2). Once more, to find the number of permutations of any number of things, taken’ four at. atime. Taking the six letters ; in the class beginning with a, that letter is followed by all the permu- tations of the other five letters taken three at a time, therefore the number of permutations of this class is, as we have seen, 5.4.3. As before, there ‘are six classes, and therefore the num- ber sought is 6.5.4.3, or 360. In like manner the number of permu- tations of m things, taken four at a time, is m.(m—1).(m — 2). (m'— 3). We may carry on this process, step by step, as far as we please. From the instances already given, however, we may conclude, that the number of per- mutations of m different things, taken nm ata time, is . m.(m a 1) .00, 3). far ey. (m'— w= 1) an expression in which the blank must be filled up with all the numbers, (if any,) between m — 2 and m—n + 1. Thus, if m be six, and n four, this ex- pression gives us, as above, 6.5.4 Bs or 360. ~ For examples : the number of permu- tations of 12 things, taken 5 at a time, is 12.11.10.9.8, or 95040. .The number of permutations of the 26 letters of the alphabet, taken 6 at a time, is 26 .25.24.23,22. 21, or 165765600. 258. When 2 is made equal to m, this expression becomes m.(m—1).(m— 2)... (m— m+ 2).(m™—m + 1); or m.(m—1).(m— 2)...3.2.1; since 1 m—m+ 2 2, and so on. This, therefore, is the num- ber of permutations of m things taken m at atime; that is, the number of ways in which the whole number of things may be arranged. For example, the number of ways in which the eight letters in the word Scotland can be written is be m—m+ i) ARITHMETIC AND ALGEBRA. 8.7.6.5.4.3.2.1, or 40320. 259. The number of things remaining the same, when we increase the number of them taken at a time we increase the number of permutations. This appears from the expression art. [257]: There is, however, one exception, and that is, that the number of permutations of m things taken m— 1 at a time is the same as when they are taken m at a time. In these respective cases the ex- pression becomes mi (m—1).(m—2)..448.2, and ’ m.(m—1).(m — 2),..3.2.1, and these expressions are equal. For example, the number of permutations of 9 things taken 8 at a time is 362880, ‘and the number when they are taken 9 ata time is the same. The reason. of this is plain, Suppose that we have formed all the permutations of m letters taken m—1atatime. A certain class of these does not contain the letter a, to all these let us prefix that letter. In like manner, let us prefix the letter b to the class that does not contain it, and so on. When we have done this, we have, without increasing their number, changed the. permutations of m things taken m — 1 at a time into the permu- tations of the same things taken m at a time. ; 260. Let us now propose to find how many different combinations can be made of any number of things of which ‘a certain number are taken at a time. In the first instance, let us suppose that there are eight things, and that they are to be taken four at atime. Let us call the number of their combinations, which, as yet, we do not know, #. Now, if we take any one of these x combinations, and arrange the four things, of which it consists, in every possible way, we shall ‘form, as we have shown in art. [258], 4.3.2.1 different permutations of them. Again, if we take any other of the x combinations and treat it in the same way, we shall form of it, in like manner, 4.3.2.1 permutations; and these are all different from the former ones, since they are composed of a dif- ferent set of things. In the same way we may form 4.3.2.1 permutations of every one of the # combinations, and when we have done so, we shall have formed Hi 4i3>rBd 79 permutations in ‘all. But these are plainly all the permutations that can be made of the eight things taken four at a time; and we know [art. 257] that the number of permutations of eight things taken four at a time is | i SB. foO ups We, therefore, have eas 19) 8 ue 6. os whence 9 B27 +6+5:_ 1680 pf Tiageracpvo epgiberuhn - 261, Just in the samé way we may find the number of combinations of m things taken m at atime. Calling the number, as before, w; every one of these # combinations furnishes us with Nee lye eo erat . different permutations of the # things of which it consists, taken n at atime [art. 258]; and, therefore, the whole a com- binations furnish us with x.n.(m—1)..3.201 such permutations. As before, these are all the possible permutations of the m things taken 2 at a time, and are, therefore, in number [art. 257] m.(m—1)...(m—n+2).(m—n+1). As before, we have the equation D. Ri (= 1). 6384261 . =m.(m—1)...(m—n+2).(m—n+1); And from this we find _m.(m—1)...(m—n +2) .(m—n+ 1) POR arty tars BY : pe? or inverting the order of the factors in the denominator é mm—-1m—2 azal ® tar Pe ony ee m—-n+2 m—nt+1 fie-kh On n f Thus the number of combinations of 12 things taken 5 at atime is, by this ex pression, Pe hs The oes —— 9 ——— 6 ee 8 ee ee or 792. Observe that this expression consists of the product of a set of fractions,.the denominators consisting of the natural numbers, and each numerator formed by subtracting the denominator from m+ i. et 80 262. The expression in the last article, being the number of combinations of a certain number of things taken a certain number at atime, must always furnish us with a whole number. This also ap- pears from the nature of that expres- sion. Thus, since either m, or m— 1, m.(m — 1) ae must be a whole number. [art. 62]. Again, either m, m—1, or m — 2, must be measured by 3, and one, at least, of these numbers is even, and, therefore, 4 -(m > 1) .(m a 2) is a whole number. Similarly, either m, m—1, m—2, or m — 3, 1s measured by 4, and some one of them different from the one measured by 4 is measured by 2; also, some one of them is measured by 3, and, there- fore, their product is measured by 1.2.3.4. In the same way it can be shown, that the numerator of the general expression is measured by 1.2.3... (2 —1).n. ; 263. Suppose that we have got ten things, and proceed to form all the com- binations of those things taken seven at atime. To form one of these combina- tions we select seven of the ten things, and, consequently, reject three; the three rejected ones form one of the com- binations of the ten things, taken three at a time. In like manner every one of the combinations of the ten things taken seven at a time has corresponding to it a combination of the same things taken three at a time; and when, by selecting seven of the things in every possible way, we have formed all the combinations of them taken seven at atime, we have also rejected three of the things in every pos- sible way, and so have formed all the combinations of them taken three at a time. It follows, that the number of combinations of ten things taken seven at a time is the same as the number of ’ combinations of the same things taken three at a time. Let us see how this property follows from the expression in art. [261]. By that the number of combinations of ten things taken three at a time is 1039.3 1.2.37 and taken seven at a time is 10.9 2S ods 0 2D— A LB) 4615 9.6.27 | Now in the last of these expressions, must be an even number, ARITHMETIC AND ALGEBRA, 4.5.6.7, isa factor, both in the nume- rator and denominator, we may, there- fore, strike it out of both, and we find 10.9.8 | 2 eS" which is the first expression. _ 264. Similarly, since, in forming every one of the combinations of m things taken 2 at a time, we reject m — n of the things, it follows that the number of combinations of m things taken 2 at a time is the same as the number of com- binations of the same thing's taken m— 2 at a time. .To find the latter of these numbers, we must observe that the last factor in the numerator of the expression in art. [261] becomes, in this case, m—(m—n-+ 1), or 2+ 1,so0 that that expression becomes m.(m—1)....(m—n+1).(m—n)... dee bois n altel) te (2-2) (+ 1) (m =n =1). (=n) Here all the. numbers, from n+ 1 to m —n inclusive, are factors, both in the numerator and denominator. Striking these factors out, the expression be- comes m.(m—1)i.(m—n+ De 1. 200s n a and this is the expression for the num- ber of combinations of m things taken 2 at a time. 265. There is but one combination of m things taken m at a time, since there is but one parcel that can be formed of them all. Again, strictly speaking, there is one combination of m things taken none at a time, since there is one way of rejecting them all. Attending to these observations, we may form the following table, showing, with respect to any num- ber of things from one to ten inclusive, how many combinations can be made of that number of things, any number of them being taken at a time. 11 19.5 1 173 aio ink LA 6 ae Lo S10A0 ode A 16 1520 15 6 1 17 i235 1B) 2140.0. 1 18/28 564470: 56 28 8.94 19 36 84 126126 84 36 9 1 1 10 45 120 210 252 210 120 45 101 The first line has reference to one thing, the second to two things, the third to ARITHMETIC AND ALGEBRA. three, and so on. The first column contains the combinations when the things are taken none at a time, the second when they are taken one at a time, the third two at atime, and so on. This table is sometimes called the arith- metical triangle. The numbers con- tained in it are possessed of many re- markable properties. 266. In each line of the table the numbers equally distant from the ends are the same. When the line has refer- ence to an even number, there is a single number in the middle of it greater than any of the rest. Thus, in the middle of the sixth line we find 20, the number of combinations of six things taken three at a time; it is different from the numbers on each side of it, which are the numbers of combinations of six things taken two at atime and four at a time, and which are equal by art. [264]. When the line has re- ference to an odd number, we find a pair of equal numbers in the middle of it. Thus, in the middle of the ninth line we find 126 and 126, the numbers of combinations of nine things. taken four at a time and five at a time. All this follows directly from art. [264]. In general, if 27 stands for any even number, there will be 27 + 1 columns in the line that has reference to 27 things. Now 27+ 1 is an odd num- ber, so that there will be a middle term in the line, and this middle term will be the number of combinations of 27r things taken 7 at a time, or it will be [art. 261] 2r.(2r—1).(2r—2)...(27 —7r +1) L. 2 E 3 We , or i 27r.(2r—1).2@r—2)..(%%7+ 1) ee te a ea ha ' Again, if 27+ 1 stands for any odd number, there will be 2 7+ 2 columns in the line that has reference to 27 + 1 things, and as 27 + 2 is an even num- ber there will be no middle term in this line, but there will be two terms equi- distant from the extremities, and equal to each other, and each of these will re- present the number of combinations of 27+ 1 things taken 7 at atime, and taken 7+ 1 atatime. The’ expression for ee first of these numbers is [art. 261 (2r+1).20.(27—1)...(227+1—r4+1) Tate Doicid bch BF Tityae 81 or ; ; (Qr+1).27r.(2r—1)...74+ 2) TOM 27 204 S090 71 1 CE The expression for the other number is found by inserting 7+ 1 at the end of the decreasing factors in the numerator, and r + 1 at the end of the increasing factors of the denominator. These two factors destroy each other and leave the expression as it was. 267. The p” number of the m** line in the table is the number of combina- tions of m things taken p — 1 at a time. Thus, the fourth number in the eighth line is 56, the number of combinations of 8 things taken 3 ata time. Now the number of combinations of m things taken p — 1 at a time is found by mul- tiplying the number of combinations of m things taken p—2 at a time by ee It follows that the p% number in the m line is formed by multiplying the number before it by _ 2 creas Thus 210, the seventh num- ber in the tenth line, is formed by mul- tiplying 252, the number before it, by 1097 at 2 b o a sn os or Dy e 268. The pg —1% number in the —— m — 1th is, as we have seen, (m—1) (m—2)...(m — p+ 2) 1 2 (p — 2) and the p” number in the same line is (m—1).(m—2)..(m—pt 2). 1 2 (p — 2) (nm — p Fl) ? CATS lig 5 ac > The sum of these two expressions is (m—1)....(m—p+2) Peer U0, hes a (9 — 2c p-l 1 or (m—T)...(m =p -- 2) m ee (2) eeqe or, removing the factor m from the right of the numerator to its left, m.(m—1)..(m—p + 2) 1a $4.2 (p —1) This last expression is the p” number in the m* line; which is thus the sum of G 82 the p—1 and p» numbers in the m — 1 lime. In other words, any num- ber in the table is the sum of the number above it, and the number above it and next to the left. Thus 20, the fourth number in the sixth line, is equal to 10 + 10. Again, for the same reason 10 in the fourth column is equal to 6 + 4, and 4 in the fourth column is equal to 3 + 1; so that 20 in the fourth column is equal to 10+6+3+1, the sum of all the numbers above it in the column next to the left. The same is manifestly true for any other number in the table. 269. In this table let us write the ver- rae columns horizontally, so that we ave PRE 2 Lo ee Seger a SG 136 1015.21 I 4 10 20 35 56 5 15 35 70 126 Here the second line contains the natu- ral numbers ; the third line the first order of figurate numbers ; the fourth line the second order of figurate num- bers ; the fifth line the third order of Jigurate numbers ; and so on. The property proved in art. [268] is there- fore this, that the m number of the p‘* order of figurate numbers is equal to the sum of the first figurate numbers of the p — 1” order. 270. Observe, that all of these expres- sions for the permutations and combina- tions of any number of things are true, only on the supposition that no two of the things in question are thesame. If two or more of them are the same, that makes a material difference in the re- sult. Thus, let us inquire in how many ways the seven letters of the word Bar- bara may be written. Let us suppose that there are seven blank spaces arranged in a line, and that we first file three of these, then other three, and so on, with the three letters a@ in the word Barbara. The number of ways in which we can do this is plainly the number of combinations of seven things taken three Re AS j ns, Aras When three of the seven blanks are filled, there remain four ; let us fill two of these with the two letters 5, this can ‘ Rae Tt) be done in ainet 6 ways. Now for at a time, that is » or 35 ways. ARITHMETIC AND ALGEBRA. every one of the 35 ways in which three of the seven blanks can be filled with the three letters a, we can fill two of the four remaining blanks in 6 ways with the two letters 6. We can, there- fore, fill five of the seven blanks with the three letters a, and the two letters 6b in 6 x 35, or 210 ways. And for every one of these 210 ways we can fill the two remaining blanks with the two letters 7. The number sought is, there- ea de. eo, sid, If all Lemet.clatehas the things had been different the number of these permutations would have been 5040. In the same way if we have m things, whereof 7 are alike, 7’ alike, 7” alike, and so on, the number of ways in which they can be arranged is m.(m—1).(m — 2) eee. P; hb Sia | TF 9 0 BY ST. 2a ee Thus the number of ways in which the eleven letters in the word Mississippt can be written is LE.160.9 867.6. 544035 i228 2.9 £7 22ST a iss or 34,650 instead of 39,916,800, which it would have been had the letters been different. We have here supposed all the letters taken together, and there is, therefore, clearly only one way of combining them. Had we proposed to find the number of permutations of m things, of which p are alike, taken 2 at a time, we should have proceeded somewhat differently. We should have separated the com- putation by finding, Ist, the number of permutations taken 7 at a time, includ- ing zone of the p things which are alike; 2dly, the number involving one of the p things, and so on. The sum of all these is evidently equal to the number of permutations required. The reader will find no difficulty in the computation of the several parts referring to art. [264] and the commencement of the present article. Thus, take the term where q of the like quantities enter. Proceeding similarly as before, let us suppose that we have 2 blank spaces, q of which are to be filled by the things which are alike, and the rest 2 —q, by the things which are unlike. Now, by the same reasoning as that made use of at the beginning of this article, it ap- pears immediately, that q of the spaces may be occupied by like things in fore, 210, or > ARITHMETIC AND ALGEBRA. n(n—1j 2... (i —g + 1h 1.2 a q different ways. Again, there arem — p things which are unlike, and the 2 — q vacant spaces may be filled by the unlike things in as many different ways as there are permutations of m— p things taken m—q at a time, or art. [264] in (m— p) (m—p—1)...(m—p— (7 —q)+ 1) different ways. Hence, since for every arrangement of the like things there are (m — p) (m— p—1).. (m—p—(n—q)+1) arrangements of the unlike, and there are n(nm—1).... (2 cg! Man 1) Ll} Ww’: oe q different arrangements of the like things, it follows, that the whole number of dif- ferent arrangements is n(n—1)..-.(m—qt]1) e Ligand 6 qd (m—p)(m—p—1)..(m—p—(n—q)+ 1). We will apply this to one example. What is the number of permutations of 20 things of which 18 are alike taken 5 at a time ? It is evident, in the first place, that at least 3 of the like things must enter into each permutation. There are then 3 sets of arrangements. Ist, Where none but like things enter, and there is evi- dently only one way of arranging these. 2dly, Where there are 4 like things, and one of the unlike. The 4 like can 5.4.3.2 1.2.3.4 ways (z= 5 and q = 4 in the expression nm(n—l1)...m—qtl i. 2 P the 2 unlike can be joined with them ; so that we have 2 xX 5, or 10, for this set of arrangements. 3dly, Where there are 3 of the like quantities, and the 2 unlike. To find this we have only to put in the general expression n=5, G=3, m= 20, and it becomes 5.4.3 1.2.3 Hence, adding the permutations in the 3 sets of arrangements, we have 31 for our result. Had we been finding the number of combinations in the same case, the pro- cess would have been similar; taking the number of combinations of the un- be arranged , or 5 different ), and either of p= 18, et; Or ou. 83 like things, instead of their permuta- tions, and considering that whatever number of like things enter, there is only one way of combining them. In the same way, separating them into different sets of arrangements, we may find the number of permutations of m things, of which p are of one kind, q of another, and so on taken ata time. Of the Binomial Theorem. 271. It will be necessary now to ex- tend the definition of a coefficient, which was confined in art. [9] to mean the numerical factor in any product. When we have a product, and are, for the time, considering any factor or factors in it as its principal and distinguishing part, we call its other factor, or factors, the coefficient of that part. Thus, if LY we have the product m . : TO am 2 and are considering it chiefly with re- a? is called the m—1 ference to 2, m 57 coefficient of w”-2, Again, if we are considering it with reference to a, — | co a”-? may be called the co- efficient of a®; and if with reference to m—1, ‘ a and 2, m nea the coefficient of a? x™-2, So that a coefficient may consist of more than one term; thus in the expression 3 (a + b + ¢) 2, 3(a+ 6 + oc) Is the coefficient of 22. 272. An expression, such as a+ 8, or 1 — x, that consists of two terms is called a binomial expression. An ex- pression that consists of three or more terms, is, sometimes, called a polyno- mial expression. The binomial theorem is the algebraical rule, or formula, for expressing any power, or root, of a binomial expression in a Series con- sisting of single terms. 273. If we wish to raise a binomial expression to any power indicated by a whole number, we can do so only by the rules of multiplication. In this way we have found [art. 176] that (w+azHatt+2anr+ a’, and [art. 181] that (7+ aps =a +3aa2+ 3a°x + ab, Again, if we take the example in art. G2 84 [29], and for each of the quantities b,c, and d there, substitute a, the ex- pression (w+a).(«+6).(#+c).(x+d) becomes x + a repeated four times, cr (x + a)*; also the coefficient of 2 in the product becomes a repeated four times, or 4a, the coefficient of 2? be- comes a? repeated six times or 6a2, the coefficient of a becomes 4a, and the term a bcd becomes a‘; so that we find (et + a)* g™ i Oto de 4...) aad ARITHMETIC AND ALGEBRA. =xz++4ax°+ 6ara?+4asan + at, In like manner if we form the product of five or six binomial factors we may deduce from it the expression for (w+ a), or (x + a)§; and if we can discover the general law according to which the continued product of m bino- mial factors is formed, we can deduce from it the general expression for (@ + a)". 274. Let us take the expression + (abt+tactad+...tbe+bd...+cd+...t¢de+...)un-2 | + (abec+abd+...tacd+...tbcd+...+-cde+...)a-3} [A]. + (abcd+abce+... tacde+...+ bedet+...)urn4 AN Reet +abcde... Here we suppose that the quantities a,b,c,d, &c. are in number m; and that the coefficient of x~1 is the sum of all these quantities, the coefficient of a”-* the sum of all the products of every two of them, the coefficient of «2-3 the sum of all the products of every three of them; so that if there were a term containing x”-", its coeffi- cient would be the sum of all the pro- ducts of every r of the quantities. This being understood, it is easy to see how the blanks in our expression are to be filled up when any particular value is given to m. Now if in this expression we make m equal to 3, we find the same result as was found in art. [29] for the continued product of three binomial factors. If we make m equal to 4, the expression becomes the same as was found in the same article for the continued product of four binomial factors. Similarly, if the reader will, by multiplication, form for himself the products of five and six binomial factors, he will find that the results are, respectively, the same as if he had substituted the numbers 5 and 6 for m in the expression above, and so for any number of factors. We may, therefore, conclude, that the expression [A] truly expresses the continued pro- duct of m binomial factors. It may be objected to this conclusion, that we have not proved it, but have raised a presumption only in favour of it, by showing that it is true in a num- ber of particular cases, and many in- stances may be brought of the fal- laciousness of such reasoning. We may, however, make our proof quite strict as follows. Let us multiply the expression [A] by # + 7,7 not being one of the quantities a, b,c, d, &e. already found in it. To do this we multiply every term in the expression by az, and also every term by 7. «a™ multiplied by w#, becomes a”, which is therefore a term in the product. a” multiplied by 7 becomes 7 #”, and the second term of [A] multiplied by # becomes (a+ 6+¢+...)a"; thesum of these, or (a+ b+c+...+ 2) 4”, is therefore another term in the product. Again, the second term of [A] mul- tiplied by 7 becomes (at + bi+ct+..)a™-, and the third term multiplied by x be- comes (@b +ac+ad+..)a™~1; the sum of these therefore, or (ab+ac+tad +... tati+be+ bd+...+b71+..) is a term in the product. Similarly x™-* multiplied by the sum of all the products of the quantities a, b, c,d... 4, taken three at a time, is another term, and so on; the last term in the product becoming a, b,c,d...7. It follows that the product of the expression [A] by x -+ 21s of the same form as[A]; m+1 being substituted for m, and the quan- tities a, b, c,d... having? placed among them. Therefore if [A] truly expresses the continued product of m binomial factors, an expression similar to [A] will truly express the continued product of m-+1 binomial factors. Now we have seen [art. 29] that [A] truly ex- presses the product of four factors, it therefore, by what we have just proved, ARITHMETIC AND ALGEBRA, truly expresses the continued product of 4+1, or five factors. Again, since it truly expresses the product of five factors, it truly expresses the product of 5 + 1, or six factors. In like manner it truly expresses the product of seven, eight, or nine factors, and so, counting up- wards, of any number of factors. It therefore truly expresses the product of m factors. Owing to the great importance of the subject, we will show that this must be the case from somewhat different con- siderations. Glancing at the products (x +a) (x+ 5b), and (x +a) (@+ 0) (@ + c) as we have written them, @tartadb, e+axnttabet+abe Oe +b2%+acz +cxu2+ bc, it is clear, from the nature of the pro- cess, that if we multiply together m fac- tors of the form (7+ a)(@+ B).... (x + h), the first term of the result will be simply the product of all the first terms of the binomials, and the last sim- ply the product of all their second terms; that is, the first term will be 2”, and the last ab...h. It is also clear that every intermediate power of # will ap- pear in the result. Again, no numbers will enter into the coefficients, since the multiplication cannot produce 2 terms exactly alike. Ifthen A, B, &c. be the sums‘of all the coefficients of "71, v”~*, &c. in the result of the multiplication as above written, we shall have it repre- sented by em +A wv" '4+Bar"-?, &e... tabe..h. Now it follows from the multiplication that, xz”! being considered composed of m —1 factors and so on, every term in the above product must have exactly m factors. Hence the coefficients of v”~’, of which A is the sum, have each of them but one factor, and that is a, or 0, or c, &c. So every coefficient of a”"~" has exactly 7 factors, and is some pro- duct arising from multiplying together r of the m quantities a, b,c...h. Now all the quantities a, b,c...h enter into the product in ‘precisely the same way, since the result would have been the same in whatever order we had multi- plied the binomial factors together. 85 Hence every possible product of 7 fac- tors out of the m quantities a, b,c... must appear in the result as the coeffi- cient of 2”—"; and we therefore see, as before, that it will be the sum of all the products of every 7 of the same quan- tities. 275. The expression [A] being thus the product of the m different binomial factors 7 +a,xv+6,x+ ¢, &e.; Jet us suppose that each of the quantities b,c, d, &e. is made equal to a. The expression (@ + a).(@ + 6).(a+).. in that case becomes x + a repeated m times, that is, it becomes (wv + a)”. As to the expression[A], its first term re- mains the same, 2”. The coefficient of the second term becomes a repeated m times, or ma; the second term there- fore becomes max”-1. The coefficient of the third term becomes a?, repeated as often as there are products of m quantities taken two at a time, that is, as often as there are combinations of m things taken two at a time [art. 261]; mm — ‘ : é 1 this coefficient is therefore m. ars : —] the third term therefore is m . a? x"-', Similarly the fourth term be- m—-1l m—2 2 haar m—-1m—2 comes m. OP , BTSs M being the number of combinations of m things taken three at atime. Proceeding in this way, the 7 +1% term has for coefficient a’ re- peated as often as there are combina- tions of m things taken r at a time, or is (art. 261] m—-1m—-2 m—r+il a6 in Salo ae et The last term of the expression [A] be- comes aaa... repeated m times, or a”. The last but one becomes a™~ repeated as often as there are combinations of m7 things taken m—1 at a time, or ma”"—a4, The last term but two be- a, SLL a" x “bed ges mm. comes ™m Qn—=2 2, Collecting these terms together we find 86 ARITHMETIC AND ALGEBRA. (B+ ay = a" + Maw + m—— : aan) + m va ht ke 2 8 a2 &e, | —lm—s§ a +m .— eae ch ack Sal 2S 2 3 +m.— zo an? 2 + mao am) + Q™, 276. The expression which we ob- tained in the last article, gives us a series of single terms together, equal to (a + x)”, when m is a whole posi- tive number, and is, therefore, [art. 272] the binomial theorem. It is also, some- times, called the developement, or ex- pansion of (% + a)”, and is of the most important use. We may observe with respect to it: First, that terms ; Secondly, that the exponent of a, added to that of x, always gives m for sum ; Thirdly, that the coefficients for dif- ferent values of m are the numbers con- tained in the table in art. [265]; and are, like them, the same for any 2 terms equidistant from the beginning and the end of the sines, so that the coefficient of 2”-? a? is the same as that of x? a™-P?, See art. [264]. Fourthly, that the coefficient of any term is formed by multiplying the co- efficient of the preceding term by the exponent of x in that term, and dividing by the number of terms preceding the one in question. This rule is of much practical utility, as it enables us to form at once the expansion of any power, without recurring to the general for- mula. The reader may satisfy himself of this by forming the coefficients in (x + a)’ by means of this rule. This law of the coefficients was discovered by Newton. Fifthly, that the coefficient of w”-? a? , or the coefficient of the (p + 1)” termis it consists of m+ 1 m (m\—1)....(m — p +1) Tee rete p The term itself is m(m—1)....(m—pt Leen t ape'+ aps p This is called the general term, because, by giving any value to p between 0 and m, it represents each particular term. And sixthly, that when m is ‘an even number, the coefficient of the middle term is m ay mm—di 2 ‘Ate a aoe 2 see art. [266], and when m is an odd number, the coefficients of the two middle terms are the same, and are [art. 266] aS m— 1 2 277. In the expression in art. [275] if for x we write unity, and for a we write y, it becomes yy = Fr ee a. m— | y? m—-1m—2 RE: Thus if for m we put y, this becomes +m y? + &e. a aoinvintg (+ ye =1+3yt3.—y 2 3 All the terms after the fourth. would contain 3 — 3 for a factor in the nu- merator, and are therefore all equal to zero, so that we have, (VEO) ee 8 8 ere 278. In what has gone before, m being in the first instance the number of bi- nomial factors in a product, is neces- sarily a whole positive number; the binomial theorem, therefore, as we have proved it, is applicable to those powers only of a binomial that are indicated by a whole positive number. We are now to show that the same theorem is true, whether m be whole or fractional, po- sitive or negative. . When m and m’ are whole positive numbers, we have seen that m—\I (+2)"=l+mze+m, 29 t-OCC. 5 and ARITHMETIC AND ALGEBRA, m —1 +2)" =1+m'2+m, 22+ &e, Now if we multiply the first members of these equations together, we tind-for the product (1+ 2)", which, by the the- orem just proved, is equal to l+(m+m)z+mt+m. m+m—1 22+ &e. Hence we have m—l1 (lt metm z+ Be. ) x (A) f—] (tmetm™ #+&e. ) = f 1+(mtm!)e+m+ m2 — Ta24 Be, The latter expression, then, is the pro- duct arising from the multiplication of the two former, and the way, therefore, in which m+! enters into the product results from the way in which m and m’ enter into its two factors. It is, there- fore, equally the product of those two former expressions, whether m and m!’ be whole numbers, or fractions, positive, or negative. Let us suppose then that we have —] #=l1+ ih le Tris 28+ &e...[X] and : = y=1-+ml 2-+ml .—— 28+ &e, »-[Y]; where m and m! are any numbers what- ever, and where we do not know any thing of w and y but that they are re- spectively equal to the sums of these two series, As before, putting p for m + m’, we have prt 2 279. Now let m be a whole positive number; in that case the expression [X] becomes the developement of (1 + 2)”; so that we have w= (1+ 2)". Also let m! become equal to — m; then p becomes m — m or zero, and the ex- pression [Z] reduces itself to ay =i, Hence we have ny=lt+pet+p a? +&e,.,{Z]. 1 Ving and, putting for # its value as above, 87 1 = ——~m, = (1 Rin ‘> ay? > ee art. [217]. Substituting for y its value in [Y] this becomes -_s dae, 2+&c.=(1+2)7™ l+m 2+m Finally, substituting for m/ its value — m, this again becomes (l+2>"™=14+(—m)z+(— m) pe be 22 + &e. Showing that the expression in art. [277] is true when m is a negative number. 230. Again, in the expressions [X] and [Y], let us suppose that m! is equal to m; this makes p= 2m, ande#=y,. Consequently wy becomes 2*, and we have, by [Z] 2m—1 e@e=1+e2met2m 22+ &e. Similarly, #2 y becomes #%, and, as be- fore, we have 3m—1 w=1+3m2+3m 2+ &e.3 and 4m —1 wt=1+4m2+4m 24+ &e3 so that in general, if 2 be any whole number, we have eee ee a =1+nmz+rnm. Now suppose that m is any fraction “ ; the last expression then becomes 1 2 a =l1+kete that is, [art. 277] Goa td ei, since kisa whole number. Taking the n‘* root of both members of this equa- tion, we find & 2=(1+2)*. For # substitute its value in [X], and this becomes a Ae ey — | mo 284 &e,=(1+2)" 5 2 l+meze+m ; * ie and finally for m substitute its yalue a 88° and we have ae eae ewok AR (1+ 2) Ben hrs, 22+ &e. S kR : L $ where | iS any fraction. This shows that the expression in art. [276] is true when m is a fractional number, and equally so whether it be positive or ne- gative. 281. Having now shown generally [arts. 276, 279, 280], that whether m be a whole or a fractional, a positive or a negative number,* we always have * The student may perhaps be desirous of knowing whether he will be justified in applying the same formula to such indexes as 4/2, 34/5, &c., which can- not be called positive or negative integers or fractions. It was shown (arts. 186, 187) that we can find frac- tions whose value shall be as near as we please to any surd quantities, And we will now show gene- rally that the binomial theorem is true wherever the index is (as in the case of surds) a quantity forming a fixed value, or limit, which can be expressed as nearly as we please, though never quite accurately, by rational arithmetical quantities; assuming only that the rule is, as proved in the text, true for these latter. Suppose m to be an index which is such a limit, and suppose / and x to be rational arithmetical quantities which express the value of m to any degree of ap- proximation, / being greater than m, and n less; so that t=m+a n=m — B, # and # being differences which can be made as small as we please. : 1.d—1) Now (1+ z) See Se gear mare BS were Ie ta ge 3 . hrs rege tees or Gey 2A ede oe Te Le m+ a.(m--a—1).(mt+a—2) , 2 8 a ee MR Rrra) ty + &c. = (A). Le —1 Let ealtmct Er) 4s Em. —1).(m—2) , pr L284. Sc. = (B). Again OPA) ahem de geek a ay -. oe n.(7—1).(” yet a oe Lot G2 83 or (1-+z)"~ =ltm—B. Fae ake os ‘ wp BOPGoe ae ; . ats 2) a mirmeecu= CO); Of the three series (A), (B), (C), the first terms are the same. | ARITHMETIC AND ALGEBRA. m—lI A+2)"=l+m2z+m, 2 + &e. we can deduce at once the general ex- pression for (vw + a)”. We have . -) < tase (1 1 Be and, therefore, _ | (@ +ay™=am (1 +<)". xv Now, as we have first seen that a a m—la? & —_ m— — a Cc. (145) oe Ga gat Cs The coefficient of the second term of (A) ism + & (B) m (C) m— 8B. Now these three, the smaller « and fare, will become more nearly equal, and they would be exactly equal if we could get rid of the differences @ and £ alto-- gether. Y ¢ The same remark applies to the three coefficients of the third terms, ” ”» 9 3° ” 9 m + %.(m-4-%—1) I é 2 m.(m—1) 1 es a ' m—£B.(m—B—1) Lee : and so on for all the rest of the coefficients. They will become more and more nearly equal, the less @ and 6 become. Now let y and 3 represent the whole dif- ferences respectively between the series (A) and (B) and (B) and (C); so that (A)=(B) + » By the diminution of # and 8, it is evident that 4 and 3 are diminished; so that the difference between (A) and (C) may be rendered as small as we please. Hence, the smaller a and £ are, the more nearly are both (1+ 2)™** and (1+z)"—* expressed by the series (B), and the approximation may be made as close as we please. But (B) will always be of inter- mediate value to (1 + 2)” and (1 4 2)™~?. Hence it follows that (Ll + 2)” =(B) exactly, For, if (1 ++ 2)” were greater than (B) by any quan- tity d, so that : a + z)™ =(B) + d, we should have (] -L z)™+# differing from (B), and still more from (1 +. 2)"~8 by a quantity necessa- rily greater than d. But this is absurd, since we may make (1 + x)", and (1-+-2)"~* approach each other as nearly as we please, by the continual diminution of a and %. And in the same way may be shown the absurdity of supposing (1 4- z)”" less than (B) by any quantity. Hence (1+ 2)"=(B)=l+m.z 2”. (m—1) m.(m—1) (m—Q)-. x8 72 4 LN aah A a ace 85 demo eB ARITHMETIC AND ALGEBRA. it follows that a (w+a)"=a (1+m—+m m— la’ ; ie aba or, multiplying each term of the series by 2”, aa” (a+ a”™=a2"+m L} - m—1 ata” + TG Cy : Bi i id or, finally, (e+ a)" = 2" + max) m— 1 +m .a m2 4- &e. This is the most general form of the binomial theorem. This theorem is generally (though not quite accurately) attributed to Sir Isaac Newton. He discovered the law of the coefficients already mentioned in art. [276]; and we refer the reader to our translation of Biot’s Life of New- ton, (p.3.) where he will find some ac- count of the manner in which Newton was led to extend the theorem to nega- tive and positive indexes. 282. It is very important to observe that the mode of proof used in articles [279] and [280] differs materially from any that we had used before. On for- mer occasions, in obtaining any result, we endeavoured to explain how it ne- cessarily followed from the nature of the question proposed, as in articles [257], [261], [275], &c. In the last articles, however, we have proved that the re- sult is true, without directly showing why it should be true; we have not ~ sought for any reason why the develope- ment of the negative and fractional powers of a binomial should be such as we haye found it to be. The first of these sorts of proof is no doubt the more satisfactory, but very often it can- not be attained ; and it is by extensively using the last, which, when rightly un- derstood, is equally conclusive, that the mathematical sciences have been so much improved in modern times, 283. If in the expression at the end of art. [281] we substitute —a for a, and observe that all the odd powers of — a are negative, while its even powers are positive, art. [30], we shall find that the alternate terms in the developement beginning with the second, become ne- gative. So that _(e@-—a=a"— man" m—1 ‘+m at g—* — &e. 89 284, We shall now apply the binomial theorem, as given in art. [281], to a few examples. To find the expansion of To find oe 1+2’ the coefficients, we must substitute — 1 for min the theorem. Now if this be that is; of (I + x)-?. done m=—1 m—1 _ (> Dy al ge ms pp yi Dig os yore oe m—lLm—2 _ ee (—1)—1(—1)—2 m. ae ae =(—1). pion ee =—] &e. So that Q+a)7!=1— a+ v— 284+ vt~ &e, which is the result in art. [46]. 285. To find the square root of 1+ a; that is, to find the expansion of (1+ me If we substitute . for m in the theorem, we find 1 m=— 2 : 1 m— 1 rae Ky | nt =-,.———_—— = ~— — 7-- 2 rae ae 2% 2 ] 1 I 9 m—-1l1 m—2 4 nO Oe m,.- ° —— 2 3 2 2 3 Sk ge yea ddl PS: ee 1 1 ] So. Beef m—lm—2m—3 12 Z Mie Faas pte ee ay ty gO es aes 2 Te Bahay. 4 &e. So that : tepid! ele Gt CA heartened 2242 «si 2" Way fe oro te" Ngee a++ &e Stas wey 04h Dd, du 4 ; 46, ] xv \2 Beas a 41 po (FS) ts (e) 1.3.5 =) 2. eet &e a aes yi 90 And, similarly, 4 1 v -() a(S). Llthipe Ft SS OER) FN a, 34 (5) -&- 2.3.4\2 2 series which proceed very regularly. In like manner we should find nts Book Ne ey atay=1->-54+55 (5) 3 ‘a. 047 if ea Buse pyointay pr, Leese. 2 1 F223 ee 8 And again, “Faith Fe Le(ey be te By Opin 1S a, Loy et Ae oy YA ——— | - ————_— | _— &e. ea a Reo 4 rae and in the same way any other root direct or inverse of any binomial quan- tity may be found. 286. The general expression for (w+ a)-" 1s gu" + (—m)an-wr.t (—m). Acai Ta. Oa or OCC 2 and this reduces itself to 1 a 4: m+ 1a? am gm 5 amt? — m m+1m+2a Mees waa This series goes on for ever, without coming to an end, the coefficient of every term being a whole number [art. 262]. If a@ be considerably less than x, the terms decrease very fast; in that case the series is said to converge, and the sum of a few of its first terms differs so m—l1 2 + &e. (1 +1)", or 2™ =1+m+%m™ And therefore am—1=mtm "1 + &e. + Now, examining the several terms of this series, and referring to art. [264], we see that they are respectively the number of combinations of m things taken 1, 2, &e. p, &c.m —1, and m at ARITHMETIC AND ALGEBRA. little from the sum of the whoie, that the other terms may be neglected. 287. The general expression for m (x + a)* is eel Eom r m Fe ae - = mM ==} m wn elle ce +— agen + — Mak pee m m ——-1—-2 4. mn n gs 2 + & ,.°? : ° a” wn C. in 2 3 7 or ibid m~—n l m=2Q0n 2 m—nN eo+e—.max”™ +—.m. aa ™ n2 2 1 m nm m4 m—~3n +—,m.—_. aa ” a 3 3 + &e. This series also goes on to an infinite number of terms, unless m be a multi- ple of », and positive, so that m= kn. In that case m — kn, or zero, becomes, after atime, a factor in the numerators of the coefficients, and from thence all the terms of the series become zero. The series then becomes of the form in art. [275], where the exponent is a whole number. Observe that we usually consider the developement of the m power of a bi- nomial to go on to infinity, without stop- ping to consider whether m be a positive integer or not; and it is clear, from the . last remark, that we introduce no error in doing so, since, in the case where the number of terms is finite, we only add a series of terms each of which is equal to zero. We shall presently find this remark of importance. 288. If we make 2 and a each =1 in the expansion of (# + a)"™,m being a whole number, we obtain ukealinstnn Meee ae ee : 2 ar uD m(m—1)..m—p+1 Let? auth lias Gaeta m—l : + &e. ty horiore t} tals a time. We, therefore, have 2" — 1 equal to the whole number of combina- tions of m things taken 1, 2, &e. up to m ata time; and if we consider the re- jection of all the things as furnishing ARITHMETIC AND ALGEBRA. 91 the one combination of m things, taken none at a time, as in art. [265], then 2” = number of combinations of m things taken 0, 1, 2,... &c. up to m at a time. This result, as well as the other, is occasionally useful in the com- putation of chances, 289. The above developement enables us to extract with rapidity the square and other roots of numbers to-any de- gree of approximation. Suppose N to represent the number, and that our ob- ject is to extract the square root of it. Take out of it the highest number which is a perfect square; call this a? and the remainder y, so that N=@+y. Extracting the square root on both sides we have WN =N@+y =aV i+ cs @ Now, art. [283], V1 +2, or(1 +2)3 Bis 1 ee =-149-5() ea +35 (§) oa CEC. 2 And putting - for 2 we get ee sie 13.3 a seh V3.7. UN 4 2.3 \2 7) If y be much smaller than a, after taking a few terms of this series, it is evident that the rest will be very small. If that be not the case, but the perfect square next below N is at some dis- tance from it, there will be a square number a little above N. Let us sup- pose 42 to be such a number, and y to be the difference between it and N. We have N=0-—y And.:../N=VB=y=b i—% Now, art. [287], 1 pha: 2 2 (l—ax)” or V/ine=1-5-5 (2) Hence putting zo for x ip es ea a a 52 962 2 \2b2 1.3 x). D) = (5 = ee “if Sash) Led fd 943 (; é2 &e,}B), Suppose N = 8. Here 9, the num- ber next above 8, is a perfect square ; we, therefore, make use of the latter series. Wehave b2=9and.'.b=3 andy =1. Hence, substituting in series B, we ob- tain Pps. 1 ] 1] 1 ‘3 2x9 2°22x 92 1 i 1 ; t 18 648 11664 Se. Now, considering all the terms in the series to be multiplied by 3, we observe that the third term, being reduced to a decimal, has no significant figure in the 2 first decimal places, and the fourth has none in the 3 first. ‘The number of terms we take note of is determined by the degree of accuracy required. See art. [166]. If we are satisfied with a result accurate to 2 places of deci- mals, we only take the 3 first terms, and have Ey Oki 8, = —— = 2.828 290. Again, suppose it were required to extract the n” root of the number N. Proceeding ina similar way we will suppose a” to be the perfect n power which is next less than N, and that N =a"+y. Extracting the 2 root, we haye "IN = "Va +y 92 1 a ree Now putting ai for m in the expansion of (1 + x)” we have —] 1 4, 1 1 — (l+a2)"=1+-—.2+—7n x? n n ee CAS Putting = for x, and substituting in the value of "WN, we obtain ly? ] PReufi — eon QO: lt And reducing and attending to the signs of the terms nfm i ln—l1 y? VNaaji4t% 187) de a VAs? n a In—12n—1y3 ers on 38n Z — koa}, Let us apply this expression to find the cube root of 31. Now, 27 being the greatest cube number contained in 31, we have a” = 27, and, therefore, a = 3, andy = 4. Hence, m being equal to 3, we have, by substituting in the value of ’ ee [ad ati io as aA) = 1 ere lame " read si=3 {145 OY eo hoger? or, performing the multiplications indi- cated, and raising the numbers to the requisite powers, 4 16 re 320 27° 2187 | 531441 291. In extracting the square root of 8 in the last article we only added 3 terms of our series, and because the fourth, when reduced to a decimal, con- tained no significant figure in the 3 first places and the terms decreased very rapidly, we concluded that the sum of °/31=3+ —&e.(A). ARITHMETIC AND ALGEBRA. all the terms after the third would not contain any significant figure in the 3 first places, so that our result was accu- rate so far.—The rapidity of diminution in this particular case rendered our conclusion unobjectionable. Where, however, the terms of a series which goes on to infinity are convergent, that is, where each term is less than the pre- ceding one, and their signs are alter- nately positive and negative, we may estimate by the following general method the error we introduce by representing the sum of the whole series by the sum of a limited number of its terms. Suppose the series to be a—-b+e—d+e—f+g—h+t &e. Let S represent the sum of the series. Now when we have taken the 4 first terms, the quantity remaining is e—ft+tg—ht+k—i+ &e, and each term being greater than that which follows it, writing this in pairs as follows (e—f) + (g—h) + (kh—D + &e we see that each of the pairs will be positive, and therefore their sum will be positive. This sum being the difference between S anda— 6 + c—d itis evi- dent that S is greater than a—d+c—d. Again, taking the 65 first terms, the re- maining ones are —f+tg-—-htk— &e, which may be written : when, since fis greater than g, h greater than k, and so on, it is evident that the whole is negative. We see then that § is equal to a—b+cec—d+e with something subtracted from it, or it is less than a—b+c—d+e, but it was proved to be greater than a — & + ¢—d; it is therefore comprised between the 4 and 5 first terms; e, there- fore, is greater than the difference be- tween f and the 4 first terms.—The same reasoning applies at whatever ~ term we stop; and we, therefore, con- clude that, in a series of the above nature, the numerical error introduced by representing the series by the sum of a limited number of its terms is less than the term which follows the last included in that sum. 292, Returning to series (A), in art. [290], and reducing the several terms to decimals, we haye ARITHMETIC AND ALGEBRA. 93 3 = 3.00000 ral = .14815 Bee er 3.14815 : 16 Subtracting [55 = 00731 3.14084 Now this series, after the two first terms, is of the nature indicated in the last. Stopping then at the 3 first terms the error is less than the fourth, or 320 531441 significant figure in the 3 first decimal places. It is clear then that .001 is greater than the error introduced by taking only 3 terms. We indicate this by saying that 3.14084 represents S31 accurately, as far as .001. The reader will find, by pursuing the same method, that °739 =°7/32+7 =2.0807, which results from taking 4 terms of the series, and is accurate as far as 0001. Should the perfect 2” power next below N be at some distance from it, so But this evidently contains no b ; that oe is not a proper fraction, or a very large one, we must take the one which is next above it, as we did in finding the square root. In that case we shall have N=a".—y o "IN leh AN CORE ey Ne 1 =a(i—+)"> a and expanding we shall be able to approximate as before.* 293. The same theorem enables us to find any power of a polynomial expres- sion. ‘Thus having (ary) =a" +many m—1 1? ee am—2 y2 +. &e., for y write a + 2, and this becomes (etatez)™ =a" tmx" (a+2z2) ot y-* (a+2)+ &e, +m expanding the terms (a + 2)°, (a + 2), &e. by the binomial theorem, this may * The series for the binomial may be represented in other forms which may sometimes be used with be converted into a series of single terms. And in like manner, if for 2 we write 6b + v, we obtain the expansion of (7 +a+6+v)", and so on. In art. [276] we found the general term of the expansion of the binomial (w+a)y", we may thus find the same for the polynomial (a + 6+¢+ &e. + A)”, m being considered a whole number. We have seen, art. [276], that in this case the coefficient of a"~”" x" in (a+ a)” m(m—1)...(m—n + 1) ; Le . 7 this was also the coefficient of a” a”-” Call this coefficient M. In the above polynomial expression suppose 6 + ¢+ &e. +e+h=a. Itbecomes (a+z)™, and the general term of this expansion is Ma” a™-”, that is, giving 2 all suc- cessive values between 0 and 2 we obtain each particular term. Call m—n, m; the general term becomes M a” #”,, and we see that 2 and m, may have any integral and posi- tive values subject to the condition that nm+m, = mM, Again, sinceeb +e+ &. + h= 2. , and that Suppose c+ &.+h= y, we have xc=bt+y, and we = (Okey) y. Let M, fat (m, — Tae (m,—p+ 1) Thor ig on nce oe 2 advantage for the purpose of approximation. We subjoin two instances, (a + x)” =a” X (<= =)" =a" (—- -) x Again, since we have n On n r % ota wy = 2", 2, f————_ 1 C—e atv 94 Then the general term of the last expan- sion is M, 6? y™-?, or M, 6? v2, ms, being equal to m — p, that is to m—n— p,orm+n + p being =m. Hence the general term of (a + 7)”, or (a+ b+ y)"is M M, a” br ys, the only condition of the 3 exponents, which must, of course, be all positive, being that their sum = m. Then making y =c + 2, and proceeding as before, and so on successively, it is clear that we shall at last arrive at the following term as the general term of the expan- SION sO War De Rie ee tte MM, M: 7. M; Ji it LP ee Egil oe aa the little letter (2) not representing an ex- ponent, but the subscript number to the letter m to which it is affixed. As before, the law of the exponents is, that, being all positive, their sum = m. m(m—1)..(m—n+1) Ng ae ~ 2 ae n M, he Ea DN ehh Je 28 Eve os p M, _ Me (my — 1).. (te = eer iy tf 2 o- q &e, = &e. And the last factor will evidently be M, Js (m; —'])..(m; —v+ I) 1.2 22 v mm, =m—nN m, =m, —p, &e. = &e. So that the last factor of the numerator of M is greater by unity than the first of the numerator of M,, and so on for all the quantities M,, M,,...M,; the product then of the numerator will be the product of all the whole numbers from m down to m —v+1. Wehave then for the general term calling mM, —v,W m(m—1)....({w+1) bas PU ok Eee e PIU ce de eg a” x bP Xe! x Ke. gr h”. Multiplying, for the sake of symmetry, the numerator and denominator by w(w—1)...1,s0 that we shall have in the numerator all the numbers from m down to 1, and reversing the order of its factors, we obtain L232 seNe Whe et m 1.2..2X1.2..p &e.X1.2.0X1.2..w a" Xx Ob K xh 2, PK AM, Again 1.2 ARITHMETIC AND ALGEBRA. This quantity will represent every term. of the developement by giving to np...wallthe values of which they are capable, subject to the above con- dition. Observe that the condition be- ing fulfilled any of them may be equal to zero, in which case the sets of factors 1. Zoe My be aa. , .0ce, belongs to those which are zero must be omitted altogether. Of Interest. 294. Interest is the value of the use of money. This value depends upon the plenty or scarcity of unemployed capital in the country, the rate of profits, and various considerations of the same nature, all, however, following from, and comprised in, the first. Estimated in this way, the value of the use of a given sum of money, though it would be a difficult problem to assign it, is evidently, the circumstances of the coun- try remaining the same, a fixed quan- tity. But the consideration paid for the use of money, which is what most wri- ters have defined interest to be, is always a matter of previous agreement between the parties to the transaction, and though necessarily dependent upon its value as above estimated is yet in- fluenced by other reasons ;—the lender considering the probability that the money will be returned, and the in- terest regularly paid, and the borrower regarding his own circumstances, and the improvement of his expectations and opportunities by the. command of a larger capital. 295. The method of settling the con- sideration to be paid for the use of any sum of money during any time is, by declaring that which is to be paid at the end of a certain time for the use of a certain sum during that time. This section will be employed in showing how the first is derived from the second. The time made use of in the present day is ordinarily a year, and the sum £100.; £5., or any sum less than £5., (the laws at present forbid alarger,) may be agreed upon and enforced as the consideration, or interest. In the case of £5. being agreed upon, the scale of remuneration, or as it is usually called the rate of in- terest, is 5 per centum per annum, or more shortly 5 per cent. Similarly had £4. been agreed upon the rate would have been 4 per cent. We might evi- ARITHMETIC AND ALGEBRA. dently have referred the rate of interest to any other period besides a year. The Greeks and Romans referred it to a month, and made interest payable (and wisely, for the reader will collect from what follows, that a short interval is desirable) monthly ; and, indeed, with us, though the rate of interest is fixed by the sum paid for the use during a ear, that sum is usually made payable in 2 equal parts half-yearly, and some- times in 4 equal parts quarterly. We may here remark, that. the sum upon which interest is charged is called the principal sum, or more shortly the principal. 296. Interest is either simple or com- pound, according to the manner in which it is calculated. Any sum being due or lent, at the end of a certain time, a year for instance, the interest upon it becomes payable, so that the sum then due, instead of being the sum originally lent, is that sum increased by the in- terest for a year. If the whole be still unpaid, and interest be still charged upon that sum only which was ori- ginally lent, and so on continually after. any number of years, then the money is said to be charged with simple interest. In this case it is clear, that the amount of interest is in proportion to the time. But if after the first year, when interest is payable and unpaid, the principal sum and interest due upon it be considered as a new,principal sum, and charged with interest accordingly, and so on continually, then the money is said to be charged with compound interest. To illustrate this briefly; £100. is due from B to A, and until payment is to be charged with 5 per cent. simple interest. At the end of the first year £5. is due for interest, at the end of the second year £5. more, and so on £5. for each year, so that the whole interest for any number of years is equal to £5. multiplied by that num- ber. But if the £100. be charged with compound interest, at the end of the first year £5. is due for interest, but at the end of the second year not £5. more, which is the interest of £100. for a year, but a larger sum, namely the interest of £105. at the same rate, and soon. — 297. The calculation of simple In- terest is sufficiently easy. From the in- terest of a hundred pounds for a year we can by a simple proportion find the interest of any other sum for the same time, and knowing the interest for a 95 year we can find ‘it for any number of » years by multiplication. Thus to find the interest of a given sum for any num- ber of years we have the following rule: Multiply the given sum by the rate of interest, divide by a hundred, and mul- tiply the result by the number of years. It is required to find the interest of £512. 10s. for 5 years at 4 per cent, per annum. Lon e siae 10> | 4 rate per cent. 100)20.50 0 20 10.00 ba pel 20 10 interest for one year. © 5 number of years. £102 10 answer. We have considered interest payable at the end of each year, we can, of course, find it for any fraction of a year by a simple proportion. Thus to find the amount of £73. 15s. for 2 years and 3 months at 34 per cent. oS, S fan 15 34 221 5 36 17 6 100)2.58 2 6 20 11.62 12 7.50 4 2.00 i. CS. : 2 11 7% interest for 1 year. 2 5 38 3. interest for 2 years, 12 10% for 3 months ———— orf of a yeiies 5 16 1% answer. 298. When a sum of money is not due for some time, but the person who owes it is willing to discharge the debt immediately, he is entitled to some com- pensation for giving up the use of the money during that time. This com- pensation is called discount. The sum paid is called the present value of the debt, and is such a sum as 96 would, at the given rate of interest, amount to the debt at the time when it is due. The discount is the difference between the debt and the present value. Now we know immediately, from the rate per cent., what £100. would amount to in the given time. And the present value would in the same time amount to the given debt. Hence the present value of a sum of money due in a given time is the fourth term of a proportion, the three first terms of which are the numbers representing in pounds the amount of £100. in the given time, 100, and the given sum. Hence we have the following rule: Multiply the given sum by 100, and divide by the number representing the amount of £100. in the given time. What is the discount on £36. 10s. due in 3 months, the rate of interest being 4 per cent. ? £100. in 3 months amounts to £101. ee ak 36.610 100 101)3650 0(36 _ 303 620 606 14 20 101)280(2 202 78 12 101)936(9 909 27 4 101)108(1 101 7 Hence the present value is £36. 2s, 93d, x. isla Sd. 36 10 0 } 36 2 92 0 7 22 answer, Examples of this kind are those which most frequently occur, and as they are worked out in the above man- ner it was thought advisable to give them so, apart from the algebraical formule, which we shall next proceed to investigate. We may here observe, ds ARITHMETIC AND ALGEBRA. that tables are in practice made use of in questions of interest and discount, by means of which they may be calculated with great rapidity. 299. Let » represent the interest of £1. for l year. (This ‘will evidently be the rate per cent. divided by 100.) And let m represent the number of years, and P the principal sum. Then the interest of £1. for 1 year being r ar nr and the interest of P for 2 years is Pm». a on eplclete cae. eects fe 2: 1s ee AGES NS, nm is Hence if I and M represent respeetively the interest, and amount of £P in the given time, we shall have the two fol- lowing equations : L=Par C1.) M=P+Par= P(1 +77) ..(Q.) The equation (1) is, under a slightly different form, the rule given in art. [292]. There being four different quan- tities in each of the above equations, by knowing any three of them we can ob- tain the other. We shall find it neces- sary in applying the equations to reduce shillings, &c. to decimals of a pound, and months, &c. to decimals of a year. Required the interest on £14. 5s, for a year and a half, at 5 per cent. By equation (1) l= Pav, INOW =e? = 32 14°55, 1 oo, m= Il15years= 1,5, TGS, Multiplying in the manner adopted in art. [167]. which becomes by reduction £10286 The interest of £73. 15s. at five per cent. amounted to £8. 5s. 112d., re- quired the time. I ro= — ew reducing I to the decimal of a pound, By equation (1) ARITHMETIC AND ALGEBRA. Av Ts 12)11.25 20) 5.9375 8.296875 P= £73. 158. = 73.75 pias 00s . Pr = 3.6875, 3.6875)8.296875(2.25 3750 92187 73750 184375 184375 And the answer therefore is 2.25 years, or 2 years and a quarter. 300. M being the amount of £P in 2 years, it is evident that P is,the present value of £M due in » years. And equation (2) of last anes gives us_ a l+ar which is, under a different form, the rule given in art. [298] for finding the present value. Similarly if D be the discount. Ai : D7 Me Men er _ Mar 6 lt+nr It is unnecessary to give examples of these expressions, as they are of the same nature as those of the last article. 301. In almost all money trans- actions, it is usual, when a deduction is made by way of discount in consequence of immediate payment, to calculate the interest of the sum to be paid, instead of the discount as above given. This -gives an advantage to the person so paying, inasmuch as he deducts the interest of the sum to be paid instead of the interest of its present value. But the person receiving is willing to forfeit the difference for being freed from all doubts and uncertainty. | In the same way interest is substi- tuted for discount in the general method of calculating equations of payments. A owes B £P, due at the end of 7 years, and £P2, due at the end of me years from the present édme; at what time must he pay B the sumof £P, and Pz, that neither party may gain or lose ? Let 2 be the number of years re- quired. Then (a —~m,) years 1s the 97 extra time during which A. has the use of P,, and he is therefore benefited by the interest of £P, for that time, or by Pi(z7—m,)7r. But he pays £Pz (72 — m) years before it is due, andis a loser, therefore, by the discount of Pz for that time, or by : P2 (m2 — n)r 1+ (1, —n)r Now, in order that he may neither gain nor lose, he must be as much a loser by paying P, before it is due, as he is a gainer by paying P, after it is due. Equating therefore his gain and loss, and dividing by r we have _ P,@ — 1) — 1t+(m, — nr’ which by reduction becomes a quadra- tic equation. But in the ordinary me- thod of treating this subject A is con- sidered a loser not by the discount, but by the interest on P, for (7, — 2) years, or by Ps (%2—n)7. In that case, we have, proceeding as before, Pi (1 —7,) = Pe(nz — 7); the solution of which gives Oe Py apes 77, Wine eee Le Siunilarly, if 2 be the equated time for the payment of any number of debts, P,,P,,P.,, &c., due at the several times N, Ne, N,, &c., we should, by the same process, arrive at the equation Pes Tee Ponte Pz mg + &e. | Pi + P, + P; + &e. . which expression is tantamount to the rule usually given: Add together the products of each debt multiplied by the time when it is due, and divide by the sum of the debis. Here, as before, the substitution of interest for discount is to the advantageof the debtor. Therule is so simple that it is unnecessary to illus- trate it by examples. 302. As soon as asum of money is payable, it matters little whether it be due under the name of principal or in-. terest; the use of it would be of equal value to its owner. It would, there- fore, appear to be equitable that it should be charged with interest in one case as well as the other; in other words, that a debt forborne should be charged with compound interest. It is, however, a singular fact that the laws of H Py(n—2n,) 98 this country never consider it so charged. Thus, supposing a person to have un- justly withheld for any number of years, the annual payment of a certain sum of money by way of interest; he would be simply compelled to pay the annual sum multiplied by the number of years. Considering interest payable yearly, the following rule for finding the amount of a sum of money at compound interest requires no explanation. Multiply the principal sum by the rate per cent., divide by a hundred, and add the prin- cipal sum. ‘This gives the amount due at the end of the first year. Multiply agatn by the rate per cent., divide by a hundred, and add the amount due at the end of the first year. The result is the amount due at the end of the second year. The same operation ts to be per- Sormed as many times as the number of years for which the interest ts to be cal- culated. The difference between the amount so found and the original sum is the com- pound interest. When there “is any fraction of a year, it is usual to add the simple interest for this portion of the amount due at the end of the preceding ear. What is the amount of £76. 15s. in 2% years at 3 per cent. compound in- terest? We shall reduce the quantities to their respective decimals, and per- form the operation as recommended in art.[167], neglecting all decimals beyond the third. cee 76.75 3 100)230.25 2.303 interest for the first year. 76.75 79.053 amount due at end of do. 3 100)237.159 2.372 interest forthe second year. 79.053 81.425 amount due at end of do. We must find the simple interest of this for three quarters of a year. Multiplying by the rate per cent. we have 244.275, and dividing by 100 2.443 This is the interest for a whole year. To find that for 3 quarters of a year, multiply by 7, or by «75, ARITHMETIC AND ALGEBRA. 2.443 75 1710 122 1.832 adding 81.425 amount due at ia of second year. 83.257 which becomes by reduction. ee Saeee B3 5s Deducting original sum 76 15 Compound interest is 6 10 14. 303. The observations made in the beginning of art. [302] apply to the calculations of discount as well as those of interest. Correctly, then, the present value of a sum of money due in a given time, is such a sum as would ata given rate of compound interest amount to the given sum in that time. The same rule applies here as to the finding the present value at simple interest; with this difference, that the amount of £100. must be calculated at compound in- terest. What is the discount on £173. 5s. due in 2 years at 5 per cent. compound interest ? Pe: The amount of £100. inl year is 105 0 110 5 or, reducing to the decimal of a pound, 110.25. Reducing also £173. 5s. to the decimal of a pound, and proceeding as in art. [167], 173.25 100 110.25)17325(157.142 11025 63000 55125 78750 7A ATG 2years . The present value is £157.142, which becomes by reduction £157. 2s. 53d., which subtracted from £173. 5s, gives for discount £18, 2s. 62d, ARITHMETIC AND ALGEBRA, 99 304. We shall retain for the alge- braical formule for compound interest the notation adopted in art. [292], sup- posing, moreover, R to represent the amount of £1. in one year, which is evidently the same as 1 + 7. Now, since £1. amounts to R in the first year, by a simple proportion R must amount to R?in the second year, and therefore £1. in two years amounts to R*. It is thence clear, that in 2 years £1. amounts to R®.. Hence we have M = PR" (1,) 1=PR"—P=P(R"-1) (2) | It is hardly necessary to observe, that these expressions present, under a slight variety of forms, the rules given in art. [302]. The following example will be sufficient. What is the amount of £5. in 3 years at 5 per cent. compound in- terest ? Here Multiplying as in art. [167], retain- ing, however, 4 decimal places, as we shall multiply by 5, art. [167], 1.05 RE T.05,and 2 = §, The amount is therefore £5. 15s. 9d. This method of calculating compound interest for any-number of years is ex- ceedingly tedious, and in practice we must have recourse to logarithms. 305. The interest having been sup- posed payable at the end of each year, it would seem impossible to calculate compound interest for a less period than a year, or to assign any but integral and positive values for 7 in the equa- tion (1) of the last article. And, in- deed, the manner in which the equation was obtained would appear to confine us to such values. But a little further consideration will convince us that, a proper signification being attached to the quantity which M represents, the expression is also true for fractional and negative values. We have before ob- served one or two instances of the ex- tension which algebraical notation gives to the terms of a question, and the con- tinuity which it implies in the quanti- ties it is employed on. The following remarks will still further illustrate this fact, while the applicability of equation 1) to all values of 7, presents another instance of the law of continuity. By the method of calculating com- pound interest adopted in art. [295], after finding the interest for one year, we added it to the principal, considered the two together as a new principal, found the interest for a year, added again, and so on. But the algebraical mode of treating the subject, consider- ing Ras the amount of £1. in one year, mentions no particular time at which the interest is to be added to the prin- cipal, to be from that time itself con- sidered as principal, and charged with interest ; on the contrary, the generality of this language forbids that any par- ticular time should be selected in pre- ference to another. ‘The conversion, therefore, of interest into principal must be considered to proceed continuously. The amount, therefore, in a year only fixes the rate of increase, and we may calculate compound interest for a less period than a year with as much pro- priety as for a greater. What is the amount of £P at com- pound interest in 6 months ? R being the amount of £1. in one year, let a be the amount of £1. in 6 months. It is evident, then, that 1, a, and R, are continued proportionals, and, therefore, at=R, 1 “.2=R’, And the amount of P in 6 months is I P R’, the expression we should have derived from equation (1), by making 1 n equal to rk Again, what is the amount of £P at H 2 100 : - 4 compound interest in the —, m part of a ear. Let x be the amount of £1. in this time. Then it is clear, from the last example, that a* is the amount in (< th parts of a year, and so on; so that a Par lg SF is a series of continued proportionals of m + 1 terms, and oe R= 1 ert. 1 “r= Rm. . 1 And the amount of P is R™, the expression we should have obtained by Ei ; putting 2 = — in equation (1). _ Similarly, to find the amount in 2 og years, and the — 7 part of a year, we have the amount at the end of 2 years equal to PAK2.7 Let this fequal \P i Aa 1 The amount of this in the — part of a 2m-+1 i L year = Fi R™ = PR2R = PR mi, Now M is the value in 7 years of £P ‘due at the present time, and in the same way as if 7 refers to a succeeding period M is the amount of £P, so if it refers to a preceding one P is the amount of £M in 7 years, _or in the latter case P= M.R’, Ag 3 seal Re Pur ae. From all this we conclude that the equation (1) of the last article ori- ginally obtained for integral is also true for fractional and negative values of 7. It will have occurred to the reader that the method formerly adopted of calcu- lating the compound interest for 23 years, where, after finding the amount at the end of 2 years, we took the sim- ple interest of this amount for 23, was incorrect, according to the principles last laid down. We shall apply these to two examples. Required the amount of £153. 10s. in 1 year and a half at 21 per cent, compound interest. R here = 1.21, and ARITHMETIC AND ALGEBRA. And reducing to decimals, the amount 3 is (153.5) x (1.2192. 3 = But {(1.21)2 = (1.21) x (1.21)? = (1.21) x (1.1) = 1.331 153.5 1.331 153.5 46 05 4 605 154 204.309 which becomes, by reduction, £204.6s.2d., which is the answer. What is the amount of £6. in 2} years at 3 per cent. compound interest ? Here R = 1.03 LS) ASE Nol cd 5 .. Re = (1.03).= (1 + 03)’. Now the following equation, art. [280], is true as far as it goes, that is, there is no term included in the &e. which is not multiplied by a_ higher power of w than the second, ee ire cece (4a4)"=14+mx+m We shall suppose x= .03 and m= 3 ? and since we only require a result accu- rate to 4 places of decimals, art. [167], we may neglect every power beyond the second of .03, for (.03)2 = .000027. We have, then, le 5 me =-— X .03 2 ARITHMETIC AND ALGEBRA, R= "1.0762 eG ree tue, 6.602, which, by reduction, becomes £6.98.24d. 306. The equation M = PR" con- tains 4 different quantities, from know- ing 3 of which we can find the other. It was observed in art. [299], that in calculating the compound interest for any number of years we were obliged to have recourse to logarithms. We are also unable to find the value of 2, when that is the unknown quantity, without similar assistance. Thus taking the logarithms of both sides of the above equation, and referring to our section on logarithms, Log. M = log. P. R” = log. P + log. R" =a los. Pree aloe. Ri. (P)? . _ _ log. M — log. P And. ae her sO Thus to find the amount of £15. 10s. in 10 years at 5 per cent. compound in- terest, we should have to multiply 1.05 by itself 10 times by the ordinary pro- cess. Referring to equation (1), and observing that £15. 10s. = £15. 5, we have and, adding, Log. P = 1.1903317 LOF) ha Sos .*. Log. M = 1.4022247 And .*. M = 25.248, which, by re- duction, becomes £25. 4s. 11d. Of Annuities. 307. From the greater complication of the subject we shall dispense with giving arithmetical rules for calculating annul- ties, and proceed at once to the alge- braical method. We thus find the amount of an annuity forborne any number of years, supposing it charged with simple interest. Let the annuity be represented by A, and supposed to be payable at the end of every year. Retaining in other re- spects the same notation as in art. [299], and observing that the interest for each year is charged on the sum of the annual payments due at the end of the preceding year, we have Due for the Ist year . A » 2nd. A oe Ar oh Srd i7cét den Ae 2 As 4 Ts oie Se eee LAS, 101 and the sum of these orz A + (1 + 2+ &e. + (n —1)) Ar gives the whole amount. The coefficient of A 7 is an arithmetic series of m — 1 terms, whose common difference is 1, and, therefore, the sum of it, art. [143], is n—1 M— 4; (2+ m=» ) 5 , OY 2. aan? n—1 *.M=nAtn, r A. = A promised to pay B £10. at the end of every year, but neglected to do so, What was due to B at the end of the 20th year, simple interest being charged at 5 per cent. ? Here A = 10,7 = .05, 2 = 20, and nm— 1 qeiate = 190, 2; and ti = 5, n—) Hence 2, FA = 95, ad -*. M = 200 + 95, and the amount due is £395, Estimated at simple interest, the pre- sent value of an annuity is found as follows. The present value of A, art. [298], to be paid at the end of 1 year, is equal to A . Pian to be paid at the end of 2 years is equal to ean and so on, and to ik: be paid at the end of 2 years = ———_-, l+nr. Now the sum of these present values is the present value of A to be paid at the end of 1, 2, and 7 years, or of an an- nuity of LA to continue m years. Thus we have P representing the present value, P = 1 1 1 A(., iy 142r hes 1+ “1. If we had supposed the first annual payment to be made at the end of the m'* instead of the 1st year, and then to continue 7 years, we should, by a like process, obtain for the present value 4+- &e. 1 1 tiem tiers 1 ir ee (7 -+ 7) + 102 What is the present value of an annuity of £5., to continue 3 years, at 14 per cent. simple interest ? Here n= 3; 1 by Mee 1+2r . 1+37)° Now r = -015 and 7? and the higher powers of r are so small as not to have any influence in the result, and may, therefore, be neglected. We have then, art. [284], Pas See . She tee } =ler+?7 Ls ee ore ae hit 7: eT eo ye: Bee ae and their sum = 3 —67 + 147%, which, putting for 7 its value, and performing the operations indicated, is equal to 2.9132, and multiplying by 5, P= 14.566, which, by reduction, becomes £14. 11s. 33d. 308. Some writers have defined the pre- sent value, estimated at simple interest, of an annuity to continue any number of years, to be that sum the amount of which would, in the given number of years, be equal to the amount of the annuity. But the sum thus obtained is not the present value of the annuity, but of the amount of the annuity after the given number of years. This amount, by n— i art. [307], is mA +n. ry A and P’ being the present value, — 1 Patan=nA+n——raA, =| , nA +. : rA or = d l+nr which differs from P the present value of the annuity, found in art. [307], as would be shown by substituting any pumber greater than unity for 2 in the values of Pand P’. The meaning we give to the expression present value would naturally lead us to expect the two quantities, P and P’, to be equal. Their inequality is the strongest proof of the inadequacy of a mode of calcula- tion, like that of simple interest, which, as it were, sets a mark upon any sums of money that may have accrued by way of interest, and forbids their future ARITHMETIC AND ALGEBRA. accumulation. The reason of their in- equality is easily explained. Suppose p to be the present value of £m due in m7 , and let us eee one year. Then p = suppose m to be unpaid for a second year and charged with interest; it amounts to m(1+ 7). But p in two years m(1+27r) l+r ’ which is different from the amount of m, and the reason is, because pr, the interest on p for the first year, is not charged with interest for the second year; and, therefore, im one case m was charged with interest and in the other only p. Therefore p, which is the pre- sent value of m, is not the present value of the amount of m after any number of years. The application of this to each payment of the annuity is mani- fest. The present: value of an annuity to continue for ever is found by making 2 infinite in the expressions for P and P’. The first becomes equal to eae ere rar t &.), a(— teres the series being continued ad infinitum ; and the latter becomes itself infinite, which is an additional proof of the in- applicability to practice of the principles upon which it rests. 309. R being, as before, the amount of £1. in one year, we thus find the amount of an annuity, forborne any number of years, at compound interest. Observing that the whole sum due at the end of each successive year is one of the annual payments, together with the amount in one year of the sum due at the end of the preceding year, we have due at the end of amounts to p(1 + 27), or to Ist year, A Qnd.. AtrAR 8rd... A+AR+AR no... A+AR+&e.+AR™, or the amount in 2 yearsis A (1+ R+ &e. + R"7!). Now the quantity within the brackets is a geometric series of 2 terms, commencing with unity and hay- ing R for a common ratio, and therefore R"—1 the sum of it fart. 151] 1 ; art. FS wee ARITHMETIC AND ALGEBRA. R* — 1 and . -M=Az_ oe Chl To find the present value estimated at compound interest we have, art. [299], the present value of A to be paid at the end of 1 year is equal to 2, to be paid at the end of 2 years is equal to as and R?’ to be paid at the end of” years is equal to A Re so that the sum of these present values, which is the present value of the annuity to continue 7 years, is Pups ke A R “+f et SG, + Ra > or, P being the present value, p= i 1+ z. + &e. + : 2 = R ; R C. Rei eee (2). The quantity within the brackets is a geometric series of 2 terms, whose com- PGT mon ratio is R and first term unity, and PP cata its sum, art. [151], is . 1 (1 - x) R-1 The present value of the amount M and.°.P=A. of the annuity in 7 years is —, or (put- bhi ting for M its value in the present article) ae R* which is the R-1 same as the present value of the an- nuity. The reader remembers that the two were different when calculated by simple interest. Had the first payment been made at the end of the m instead of the first year, and continued 2 years afterwards, we should have had by the same pro- cess the present value represented by A aX Re? Rew ] it becomes A xX + &e. + ree? or by 1038. Auf 1 1 Re ! + Rt &. “1 ea} that is summing the geometric series, art. [151], by A R” or Het Tee he If the annuity be supposed to con- tinue for ever, the series in equation (2) will go on ad infinitum, and P being the present value, we shall have A 1 1 P =5 (1 +5 TR: + &e. ad infinitum), or, art. [153], and similarly if the first payment be made after the m year, and then con- tinue for ever, the present value is A U be. Reet oR 1 The rate of interest is 33 per cent., what sum of money is equivalent to an income of £3. a year? 32 is equal to 3.4 per cent., and, therefore, R = 1.034 .*.R —1=.034. By equation (4) of this article 16 Ac inliae, #4 .8008 Miia? Perse 7 34 which, by division, is equal to 883%, or 88: nearly. The answer then is £883. 4 > Indeterminate Equations. 310. In our sections on simple and quadratic equations, art. [108], &c. and art. [204], &¢c., we considered those cases only where there was the same number of independent equations as of unknown quantities. The method of solution adopted in this case was, when there were several unknown quantities, by combining the equations in any man- ner to obtain an equation involving only one unknown quantity, see art.(119], and 104 art. [213]. We now propose to exa- mine those cases where we have a less number of equations than of unknown quantities. Imperfect as our general means. of solution were in the former case, except where our equations were all of the first degree, we shall find our powers in this case confined within still narrower limits. We begin, as before, with equations of the first degree, and first with the most simple case of one equation in- volving two unknown quantities, which may be represented generally by ax + by =c. We observe, in the first place, that if the values of x and y which we are seeking for may be of any kind, positive or negative, whole numbers or fractions, there are an infinite number of such values which satisfy this equation, see art. [119]. For we have only to sub- stitute any value for one of the unknown quantities, y for instance, and then solve the equation with respect to the other, a, that is, considering x as the only un- known quantity, and the assumed value for y, and the value found for a, pre- sent us with a solution of the equation. It is for this reason, that equations of this kind are called indeterminate, be- cause they do not fix, or determine, the values of the unknown quantities. All systems of equations, where there are more unknown quantities than inde- pendent equations, are indeterminate in the same sense of the word; for, retain- ing as many unknown quantities as there are equations, we may give the rest any values we please, and thus arrive at an infinite number of solutions. But re- turning to the equation ax + by=c, suppose that we were seeking only such values of 2 and y as being whole num- bers, or being whole numbers and posi- tive, satisfied this equation, the above method would not be of any service, for although we might assume a whole number for y, yet the value of a ob- tained in the above manner, and which with the assumed value for y would form a solution of the equation, would, in all probability, be a fraction. In the second place, the equation ax+by=ce, being cleared of frac- tions, and in its lowest terms, in order that any integral values of a and y may be capable of satisfying this equation, it is necessary that a and 6 be prime to each other. For, supposing for a mo- ment this not to be the case, so that a and 6 haying a common divisor, sup- ARITHMETIC AND ALGEBRA. pose 7, may be written under the form er and fr, where e, f, and 7 are whole — numbers, substituting their values of a and 6 in the equationax+ by =e, it becomes era+ fry =c, or, dividing by 7, ex+ fy = = But the equation ~having been previously in its lowest terms, = must be a fraction, and it is in that case evident, that the equation CLE Ty = es cannot be satisfied by integral values of x and y. We shall now show how we may find the whole numbers which satisfy the equation ax + by =c, observing that questions which give rise to equations of this kind, usually require integral values of the unknown quantities. It will be best to commence by taking a numerical example of this equation, and to consider the more general case after- wards. 311. It is required to find the inte- eral values of # and y in the equation 5x+7y = 81. We observe here, that the conditions above alluded to are satisfied, the equation being in its lowest terms, and 5 and 7 being prime to each other. Transforming to the right-hand side of the equation, the term having the largest coefficient, we have S2=81l—7y. Dividing by5, wet 81 zr vy F J Dividing out as much as possible, ] : ex©=16+—-y Aa | 4 ¢ 5 2y-1 = 116 eet) a As Ui) Our having transformed the term with the larger coefficient, has, we see, enabled us to simplify the expression by dividing ont by the smaller coefficient. Now any value of y being substituted in this equation, that value of y, toge- ther with the value of a derived from the same equation, form, in the general sense of the word, a solution of the equation. But we are only seeking integral values of 2 and y; we shall, therefore, now find such a value of y as, being itself a whole number, will, when substituted in equation (1), make the ARITHMETIC AND ALGEBRA. value of x derived therefrom a whole number. Now, in order that 2 may be a whole fry 2y—-1 number, it is necessary that A be a whole number. 2y-1 Let ue th J v being any whole number, oe SY - L= dv, P This is an equation of the same kind as the original one, but its terms are simpler, and necessarily so, from the operation of dividing out, before made use of. In order to find the values of y and v we proceed as before. We thus obtain Now y being a whole number, 2 must be a whole number. vp+l Let z ol = WU; w being any whole number. We obtain from this, Ce 2 ea Dee wes CO): From all these operations we see that, w being any whole number, the value of v derived from equation (3) is inte- gral, as also the value of y from equa- tion (2), and that of a from equation (1). The equation (3) is of the same nature as equation (1), but the coeffi- cient of » being unity we may assume any value for w, and are certain of an integral value for v. Although the coefficient of v might have been some whole number greater than unity, yet by pursuing the same process we are sure of arriving ultimately at an equa- tion of this form, the operation of dividing out constantly diminishing the coefticients of the quantities y, v, &c. Substituting the value of v derived from equation (3), in equation (2) we have y=4w-2+, or (4). And again, substituting this value of y in equation (1), 2=16-5w+2—2wv +1, or Pil — FW. eres (5). The corresponding values of x and y 105 obtained by giving all integral values to w in the equations (4) and (5), present so many solutions of the equation. The quantity w is called an indeter- minate quantity, or, more shortly, an indeterminate. Supposing then w equal to 1, we haye v= 12, y= 3. Supposing w equal to 2, | ; L=5, y= 8. And giving to w all succeeding inte- gral values from 1 upwards, we obtain the following corresponding values of rand y. i2eande. 3; Bors Tae, een eee, GPL OS pe Qs we LO, &e. &e. We shall presently return to the law which the several values of a and y follow. 312. In solving equations of this kind we always endeavour to arrive at values of w and y expressed in terms of some indeterminate w, which is susceptible of all integral values, as in equations (4) and (5). The reader will find by-trial, that unless the coefficients of x and y are prime to each other, the attainment of this result is impracticable, art. [308]. It is frequently much shortened by the employment of various artifices for which no general rule can be given. They can be only learnt by observation and practice. Thus taking the equation ll2—l7y=5. Proceeding as before, llx=17y+5, and ea 2 1] ta 6 ese 5 = ay as If we continued to proceed as before, we should put eueare? =v, but ob- serving, that the difference between 6, the coefficient of y, and the denomina- tor 11, is equal to the other term 5, we put the above equation under another form, namely, 106 Tly—5by +5 , : - 1 Now 5 being prime to 11, y must be a whole number. ot conceited = WU, us 11 44-1 = 11, and go 11 eae Also w=2y—-5w, =22w+2—5u, or x= 17w-+ 2. So that the several integral values of x and y, obtained by giving. to w all in- tegral values from 0 upwards, are 2 and 1, eee BG bay «1 2s &e. &e. To find a number which, when divided by 6, gives aremainder 5, and when divided by 7, a remainder 3. Let us suppose 6 to be contained & times in the number in question, with a remainder 5, then the number is 6% + 5. And similarly, supposing 7 to be con- tained y times, with a remainder 3, the number is7y¥ +3. Equating the two values of the number, we have the fol- lowing equation: 6775 = /y + 2, or 62%=7y — 2, in 6 i ae 2 6 pe y-2 ae, ; - 2 Let Y = W, 6 then y=6w + 2, and LC=6wW+2+4, =7w+2 The several pairs of values of a and y are 2 and 2, Dodge ele Ge Ve it. Bote. 20, &e. &e, ARITHMETIC AND ALGEBRA. And the several values ‘of 6a + 5, or. 7y + 3, or of the number required, are 17, 59, 101, &e. numbers which will be found upon trial to satisfy the proposed condition. A person has only crowns and 3 shil- ling pieces in his pocket, and wishes to. pay a bill of £2. 16s. How many must he give of each? Let 2 = the number of 3 shilling pieces, AIC Y = ale te! whe crowns, then the sum expressed in shillings is 3a + 5y, and therefore we have by the question, 32+ 5y = 56, 3X =56 —5Y, D0 Peo = ay —~ 2 or ew=18 -y- d : 3 | 2(y — 1) - 1 Let 2" =», 3 then y¥-1=34,, and y= 3wt+l1; also ®%=18 —- (3w+1)—2w, =17-5w. And the several pairs of values for x and y are 17 “and 4 1 See ee tax agohied Ges) pabee. , = B and 13, &e. &e. So that 17 pieces of 3 shillings and 1 crown, 12 pieces of 3 shillings and 4 crowns, &c., make up the sum re- quired. The negative values for a sig- nify that so many pieces of 3 shillings must be given back. Thus the sum may be made up by giving 13 crowns, 3 pieces of 3 shillings being returned. Suppose it had been required to pay the same sum in crowns and half sovereigns. A moment’s consideration shows this to be impossible, and forming the equation we shall find that the coefficients of x and y admit a divisor 5, to which 56, | the number on the other side of the equation, is prime, art. [310]. 313. If we observe the several values of either of the unknown quantities in any of the above examples, we shall find that they form an arithmetical progres-: ARITHMETIC AND ALGEBRA. sion, the common difference of which is the coefficient of the other quantity. For instance, in the first example, 5a@+7y = 81, the several values of x were 12, 5, a 2, beat! 9, &e. and those of y g Sb6 13> 418,18. Tn the former we observe each term is less than the preceding by 7, the coefficient of y, and in the latter, each term is greater by 5, the coefficient of w. In order to show that this is neces- sarily the case, we will consider the general equation aw+by=c. This equation is in its lowest terms, and @ and 6 are prime toeach other. Suppose we have found 2 and y/, two integral values of @ and y, which satisfy this equation, we have Ga! + Dy = Cum «i 1), Now let x +m, and y' +n, be two other integral values of x and y, so that a(a’+m)+ b(y’+n)=c..- (2). We propose to find what relations must subsist between m and 7. Substracting equation (1), term by term, from equation (2), we have am+bn=0, am= — bn, Ci Tay heey: therefore m=-— reat Now observing that 6 and a are prime to each other, in order that m may be a whole number it is necessary that 7 should be a multiple ofa; and 2 must also be a whole number. Let then n= aw, w being capable of all integral values, positive as well as negative. Equation (3) gives usm = bw. Hence a’ and y! being any two values of 2 and y, we have all others represented by a’ — bw, yaw, w being capable of all integral values. From this result we learn, /irsi, that the several integral values of x and y, which satisfy the equationaw + by =c, must necessarily be of the form above indicated. Secondly. That a and 6 being both positive, while the value of y is increased by the coefficient of z, that of x is diminished by the coefficient of y, and vice versaé ; but if aor b be negative, then their corresponding values are both in- creased or both diminished by the coefficient of the other, or 107 Thirdly. That the indeterminate w being capable of negative as well as positive values, the arithmetic series at the beginning of the present article giving the several values of x and y, may be continued indefinitely to the left as well as the right. It is clear from what has preceded, that having obtained any two integral values of # and y, which satisfy the equation, we can immediately (from the equations v=2' —-bw, y =y' +aw) find an infinite number of such values. The main object then is the finding with ra- pidity two such values. A property of a continued fraction has been made use of for this purpose, and we shall briefly explain the manner in treating of that subject. 314. When we have one equation of the first degree involving more than two unknown quantities, the method of pro- ceeding is very similar. We shall simply go through the operations, their expla- nation being the same as that given in the case of two unknown quantities. 4x+9y+ 10z2= 103, transforming 42=103 = 9y — 102, 103 ~9y— 102: and app oi ak Neco 4 ? +22-3 —3 Let yt it =u, Ny t%2-3=4, and. y=4w+3 —2z2....(2). Substituting this value of y in equa- tion (1), e= 25 —-8w-6+42-—-22--w, OF a= 19 — Foe Ver es): We have thus the values of x and y expressed in terms of 2, one of the un- known quantities, and an indeterminate w, but the quantity 2 is so involved that giving to it any integral value, the cor- responding values of a and y are also integral. Suppose z equal to 0, and giv- ing to w sucessively the values 0, 1, 2, &c., we find the corresponding values of x and y to be 19 and 3, SGPT cs: Fs he ee 1h &e, &e. Next suppose 2 = 1, the corresponding values of x andy are 108 21 and 1, te Dee ST aAa se Fy SBA iy ALR and so on. Ifit be necessary that the values of all the unknown quantities be positive, we must not give to z a greater value than 9.. In that case x and y are each equal to unity. A similar method may be adopted whatever be the number of unknown quantities, and we shall ultimately arrive at two values of a and y, expressed in terms of the other unknown quantities, and one indeterminate, or we shall have the values of several of the unknown quantities expressed in terms of the others, and of several indeterminates. 315. Still considering only simple equations, suppose that we have several equations involving, however, a greater number of unknown quantities, as, for example, two equations involving three. de2+1lly+9z=360....(1), wtytz2=30....... (2). Our object is to find such ¢ztegral values of x, y, and z as satisfy at the same time both these equations. The process adopted is analogous to that in art. [119], where we had two equations between two unknown quan- tities, and to the reasoning of that article the reader is referred for an explanation of what follows. Multiplying equation (2) by 14, the coefficient of x in equation (1), we have 14” +14y + 14 2 = 420, Subtracting equation (1), term by term, from this, 14 # disappears, and we have 3y +52 = 60 (3). From this equation we find the values of y and 2 in terms of an indeterminate zw, which values are as follows, y=5+5W z2='9 -—Sw (Gus Subtracting these values of y and z in equation (2), and reducing, we obtain w=16-2w (6). Making w then successively equal to 0,1,2, &c.,; we obtain the following cor- responding values of x, y, and z: eee eerer eee oe ef see we eo 6 @ 16, 5 and 9, TiVO Lee 6, Me. ld Be '3, 10 500s 19, &e. &e. ARITHMETIC AND ALGEBRA. In the above equation the coefficients of the unknown quantities in one equa- tion being equal to unity, the process was very simple. The following example will show more fully the method of solving equations of this kind. 20. Yer 3s = 1083S, 3 ee a we 95 oe eae Multiplying equation (1) by 3, the coefficient of x in equation (2), and equation (2) by 2, the coefficient of # in equation (1), we obtain bat 1lby +92 ='324) 6% —4y + 142 = 190, and, subtracting, 19 y —- D2 184 Oe ees (3). The values of y and z obtained from this equation in terms of an indeterminate #, are YA BT a ae ee eae 2=15+19¢ (5). Substituting these values of y and z in equation 2, we have oe = 22 — 100 4+ 105 look toe and, transposing, 3 = 1232 Ay, Here we have again an equation between two unknown quantities, to which we should apply the method for solution of equations of this nature, and thus have values of x and ¢ in terms of another indeterminate w. The substitution of the value of ¢ in the equations (4) and (5), would give us the values of y and 2 in terms of w. We should thus have the values of 2, y, and z in terms of the same indeterminate w. But observing the above equation, we see that all its terms are divisible by 3, and, dividing, we obtain II eevee eve eee e+41t=4, which gives us at once md ~ Ae Po ee (6). Thus equations (4), (5), and (6) give us at once the values of the three un- known quantities in terms of a quantity t, to which we may give any integral value. It will frequently happen that equations of this kind do not admit a solution in whole numbers, and that will be indicated by one of the equations which we arrive at between two un- known quantities, having its coefficients divisible by a number, of which the other sum is not a multiple. See art. [310]. 316. By combining any two simple equations, whatever be the number of ARITHMETIC AND ALGEBRA. unknown quantities comprised in them, in the way in which equations (1) and (2), in the last example, were combined, we may always arrive at an equation in- volving one unknown quantity less than those two equations. See art. [119]. When we do this, we are said to elzmi- mate that unknown quantity. Thus equation (3), in the two last examples, arose from the elimination of x from the equations (1) and (2). Suppose then that we have p equations involving any greater number of unknown quantities. Combining the first of these equa- tions with the second, third, and every other successively, and eliminating by each combination the same unknown quantity, we arrive at p — 1 equations, involving one unknown quantity less than the p original equations. Thus combining the first of the p — 1 equa- tions with every other successively, we get rid of another unknown quantity, and arrive at! p — 2 equations. Proceeding similarly, we shall at last arrive at one equation, involving a certain number of the unknown quantities, which we may solve by one of the methods which we have given, and find values of the un- known quantities comprised in it in terms of one or more indeterminates. This will lead us, by the continuation of a similar process, to the values of all the unknown quantities. It might at first appear, that, after combining the first of our p equations with each of the other, we might also combine the second with the third, and so on, and thus, elimi- nating the same unknown quantity as before, obtain more independent equa- tions. But that, this is not the case may be proved as follows. Having trans- posed all the terms of each of the equa- tions to the left-hand side, art. [109], we may write them A =0, B =0,C =0, &e. Suppose a, b, c, &e. to be the coefficients of w, the quantity which we propose to eliminate, in the equations m= 0," eat Oe © = 0.) Sc) ee Thenk. in order to get rid of x from A and B, we multiply the first by 6, and the second bya, and subtract. We thus obtain Ab—-Ba@=0 .-..'%. wore CPR Similarly, combining A = 0 with C = 0, we obtain PCr — FER PRIN e's C2) Now combining B = 0 with C = 0, the equation we should arrive at, indepen- dent of 2, is Pee at Oe eee ee GON: 109 and this equation is not independent of the equations (1) and (2), but derivable from them, as follows. Multiply equa- tion (1) by c, and equation (2) by 0, and subtract, we obtain Bac-—Cab=0, or, dividing by a, Be - Cb=0, which is the same as equation(3). The Same may be proved of any combina- tions of the equations, We may hence conclude, that, when we have any num- ber of simple equations less than the number of unknown quantities, we can- not, by combining them in any manner, arrive at the same number of independ- ent equations as unknown quantities. 317. When indeterminate equations are above the first degree, their solution is one of considerable difficulty. If in one equation with two unknown quantities even the square of either of them occurs, its discussion would require a separate treatise. We will take one example, where the only term above the first de- gree is the product of the two unknown quantities, the solution of which will give some notion of the method to be generally adopted. Required two numbers whose pro- duct, added to three times the first, and five times the second, is equal to 48, x and y being the two numbers, the question gives us the following equa- tion : Ly t3xe+ sy = 48. Transposing the term not involving a, and collecting the coefticients of a, “(y+ 3) =48 —- by, and dividing by the coefficient of x, 48 — Sy “ys Se We may get rid of y from the numerator by writing this pas Boy = 5 Gir ) ie 63 de -5 or renee Now, in order that 2 may be a whole number, 63 must be a multiple of y + 3, and the several numbers of which 63 is a multiple are 3, 7,9,21. Hence y+ 3 must be equal to one of these numbers, In the first case y would be equal to 0, but if we take 7, we have y = 4, and 110 a iy ts =9-5=4, or the two numbers are 4 and 4, as will be found correct by trial. Again, taking 9, we have y = 6, and 63 and we shall find the numbers 2 and 6 answer the proposed condition. The other value of y+ 3, viz. 21, would make w negative. On Continued Fractions. 318. When we have any fraction, as ; Baa tae ee for instance 73! dividing out as far as we can, the fractional part will become a proper fraction, and we have a mixed ; 9 ‘Spe gs fraction 4 73” ot representing it alge- braically, 9 4 + 13° Now this may be written 1 4+ 3 9 We may now proceed further with the division, for 13 4 Writing this in the; denominator we have 1 4 Pa ODS ee 9 4 1 1 A in, a gar ee 4 4 And putting this value of : in the pre- vious expression, we have the fraction 61 4 v1 represented by the following expres- sion: So lee l+—4 ara ARITHMETIC AND ALGEBRA. A fraction represented in this way, where we observe the numerators of the several fractions which enter into it are all equal to unity, is called a con- tinued fraction. We shall here be able to enter into a few only, and those the most simple and obvious, of the proper- ties which it possesses. In the first place it enables us to ex- press in simpler terms the value of any given fraction to any degree of accuracy which the question may require. ‘Thus taking the above expression represent- ; ues OL ‘ ing the fraction ie and neglecting the fractional part altogether, we have for our first approximation 4, which differs from the accurate value of the fraction by 5 . Again, taking only the first term of the denominator, and neglecting the rest, our second approximation will be 4+ 7 or 5, and this differs from the - ay 4 accurate value of the fraction by vai Observe that this approximation is greater than the fraction itself, while the forrmer one was less. Proceeding one step further, and representing the fraction by . 1 ity ITs 4+ ‘ ; : : 2 our third approximation is 4 + 3 oF 14 : : ; + ieee mt which, like the first approximation, 1s less than the accurate value of the frac- , . , A tion, and differs from it by ert approach- ing nearer to it than either of the former values. We shall presently treat the subject more generally, and shall then see that it necessarily results from the form of the expression that the succes- sive approximations, derived in the above manner, err alternately by excess and deficiency, and approach nearer and nearer to the accurate value of the frac- tion. As an example of this, we will exa- mine the length of the tropical year, or of the interval between the sun’s leaying - ARITHMETIC AND ALGEBRA. and returning to the same equinox.* The most accurate calculations have proved this interval (upon which the seasons mainly depend) to be equal to 365,242264 days. Representing ‘this fractionally, we have 242264 1000600’ or, reducing the fractional part to its lowest terms, 365 + 30283 5 jE ee EN Tae, an 125000’ + 125000 30283 And, dividing as before, we obtain 1 365 ee u ges 3868 30283 This, by a further reduction, becomes 1 365 + 1 4 ———_,, + 3207 3868 Not proceeding any further with the division, we see that the three first ap- proximations to the given fraction are 7 365, 3655, 36555: The first number gives us a very rude approximation to the length of the year, and one which, in the course of a few centuries, would completely invert the order of the sea- sons. The second answers to the Julian Calendar, by which one day was inter- calated every four years ; and the third differs, by a very small quantity, from the length of the day which is the basis of the Gregorian Calendar, and which is now acted upon in almost all the countries of Europe. According to this we intercalate a single day every four years, but omit three of these inter- calations in four centuries. This makes us intercalate ninety-seven days in four centuries, and gives therefore for the 97 length of each year aD te days. 319. But the utility of continued frac- * In reality, the earth moves round the sun, but it aids conception sometimes to consider the earth at rest, and the sun moving round it. The sun is said to be in the eguinow in those two positions where the plane of the earth’s equator being pro. duced passes through him. At these times the duration of day is equal to that of night all over the earth, 3 111 tions in giving us approximations to the value of any fraction in simpler terms than the fraction itself, is not to be esti- mated very highly. Their principal utility consists in our frequently being able to express the value of an unknown quantity under this form only, or under no other so easily. On this account every thing connected with them be- comes important. For instance, when we have the quantity sought for, only under this form, it is convenient to know, in stopping at any term of the continued fraction, how far our ap- proximation differs from the value of the whole of it, or to know some limit to the error, so as to estimate the degree of accuracy. See art. [292]. We shall presently show how this limit may be assigned. In art. [254] we showed how, from an equation of the form a = b, we might find the value of # by means of a table of logarithms. We may also express the value of w derived from such an equation in the form of a continued fraction. Then take the equation Qa = 35 Now, observing that 21 = 2, and 2 = 4, it is evident that in the above equation x must be greater than 1, and less than 2. Suppose, then, “8: v=1+-, a! where 2 is greater than 1, (since 2 is less than 2.) We have from the original equation ue DB ees cme e or 2x 2” =3, 1 on 3 or pe a ee 2° or, raising both sides of the equation to the x!” power, Pay a = eae 2 Ns 3 te Now 2) — 3g? Which is less than 2; ce Ch). a \% 4G 2 but eG) ir & which is greater than 2, and therefore 2/ must be greater than 1, and less than 2, 112 Suppose, then, ; eH ae x Substituting this in equation (2), i422 SNA Goi ) = 2 : > NE or = 9 9 r) - 4 ath or Shi 3 which gives us Oe 4 y ; a8 (: is less than " sade Now and 4 3 (=) yor, is greater than ae whence we infer, as before, that @” is ereater than 1, and less than 2. Let, then, ] t = Nae. el Equation (2) gives us or or which gives us iG)" We should here find that a2” was greater than 2, and less than 3, and might again proceed in the same man- ner as before. The result of all this is, that 1 c= ] + ae or substituting for x’ t=Ae 1+; ARITHMETIC AND ALGEBRA. ' or substituting for x” or since a’ is greater than 2. The several approximations to the value of a derived from this expression are 3. 8 . 2 > 5 > 320. In order to consider this sub- ject generally, and exhibit the relations which the successive approximations bear to each other, and their respective degrees of accuracy, we will take the general form of a continued fraction, or 1 y+———_— Y ae en aaa ee vy an wT yt Be. where ¥, Yi, Yo, Y3» &e. are whole numbers. The first approximation tc the value of this fraction is y, 1 Yaig ve &e. 1, 2, the second is the third is and so on. With regard to thei several approximations, we immediately observe that the first y is too smalé there being some quantity, to be added to it; 1 Yi + &e. that the secone i Be ! y+ a is too great, the denominator 0 1 the fractional part not ee yi bu tity, some quantity, y, -- ——— A aoe , greate than y,; the third, y—-- ee ee to Hite ya small for y, -+ ER 2 the fractional part is too great for th , the denominator 0 ARITHMETIC AND ALGEBRA. 1 same reason that y +—+-— was too great I in the second approximation, and so on. It is unnecessary to proceed further to prove the following principle. The suc- cessive approximations to the value of a continued fraction are alternately too great and too small, the even ones too great, and the odd ones too small. _ 321. In order to exhibit the connec- tions of the several approximations with each other, let the /irst be represented by i the second by = the third by ug fi qn q2 Pi : 2 and so on. The several fractions —, Pa | Qo &e. are called converging fractions, or more shortly convergents Ji the first, ae 1 2 the second, &c. and their several numera- tors and denominators are intended to represent the numerators and denomina- tors of the several approximations, when reduced to the form of simple fractions by the process made use of 1n art. [318]. The” approximation thenis the (7 — 1) converging fraction ; 2 is not called a converging fraction, because, being equal to y, it is in fact not a fraction; q is of necessify equal to unity, but it is put under this form for the sake of sym- metry. The following equations need no ex- planation, — ee, qd 1 Pie 1 es ere reducing this becomes Pr vy td n Yi. so that ea: TE ie (1), n=" p2 . ay. 1 : — is found by writing y, + — for yi G2 Y2 the expression for , so that it is equal 1 to 1 a 1 et Y. Ya 113 Y pryaty + ye Yi yar il which may be written YAtrDyty Yi Ys =) : Now observing equations (1), and also that y = p and q = 1, we obtain by substitution in the above value of i q2 Pe i Pri Yor p g2 M1 Y2 a q so that P2 =P, Y27 P 9 EL ie gig? eats rae yy Q@= nyt q Ps . Sone is found by substituting ye + a . 3 p2 qa for v2 in the value of —, and is there- fore equal to sib: Pi (m +— a Pp Ys n(wte)+a xX Ys which may be written (Yet PY, TP . (hyztQyet nu and observing equations (2), and sub- stituting for p, ye + p, and q, yz + q, we obtain Ps _ P2Yst Pr WE G2 ¥3 + Q ; so that Ps = P2Y3 7 ot bat, a con (3). Gs = aye tN Proceeding similarly, it is evident from the way in which the quantities y, y,, &c. enter into the several converging fractions, that we shall arrive at a similar result, and we have the following equation connecting each of the con- verging fractions with the two which precede it : Pn fy Pn-i Yn +- Pn—-2 . dn Un--} Yn + Ges whence Pn = Pn-1 Yn + Pn-2 weenie (2): Qn = Qn-1 Yn- Gn-2 322. Multiplying the first of these equations by qn-1, and the second by Pn-1, and subtraciing, we obtain Pa X Qa-1 = Qn X Pn-1 = Pn-2 Yn-1 ~ Qn-2 Pn-1 or Pu X Qn-1 = Yu X Pun = — {Paar Gna ~ YnarPn-2b vores (A): I 114 This expression shows us that the difference between the product of the numerator of any converging fraction, and the denominator of that which pre- cedes it, and the product of the denomt- nator of the same converging fraction, and the numerator of that which pre- cedes tt, 1s alternately positive and nega- tive, and differs only in sign. Now going back to equations (1), and observing that y = p and gq =1, we haye DXG-UxXpHyntl-yy, or Pa = sp = 1. pee recurring to equations (2), we nc PoXh—- gx p=pxn-qxpr (aX G-h Xp) =—1] aresult which might have been at once inferred from equation(A). The follow- ing equation immediately results from equation (A) and the remark which follows it: 5 OX Maa On Go SP +> (BR). And it is also evident from the values of Pi Xq— X p,and ps X G1 — qe X Pr» that the positive sign must be taken when 2 is odd, and the negative one when it is even. We may hence derive the following property. The difference between the product of the numerator of then converging fraction, and the deno- minator of the(n — 1)", and the product of the denominator of the nt, and the numerator of the (n — 1)*, 7s equal to --1 when n is odd, and — 1 when n ts even. 323. In treating of indeterminate equa- tions, we observed, art. [311], that by means of a property of continued frac- tions we might quickly arrive at a solu- tion in whole numbers of the equation awv-+-by=c. We may thus find it. The fraction 5 being reduced into the —S eee — Le re Q present the converging fraction previous to the last, which will be the complete form of a continued fraction, let hate} fraction — 7 We have then by the last rule Oe ae ee tT 1. In any particular case we should know re whether a was an even or an odd con- ARITHMETIC AND ALGEBRA. verging fraction, and therefore know — which sign to take. We will suppose — it to be an odd one, and according to — the rule adopting the positive sign, and then multiplying by c, we have Oo Poe cae he EN A eo ee Cae Now observing the equation ax-+by=e, and comparing it with equation (1), we see that it is satisfied by the substitution of P x efor x, and Q x cfory. Hence P xc, and — Q x ¢, afford us a solution of the equation. By applying this to the equation we before examined, 5 @-++7 y = 81, we should obtain for the values of wand y, 243, and — 162. Returning to equation (B), Pn X Qn—-1 — Qn X Pn-1 etl, it is clear from this equation that every divisor of p, and gn must also divide 1, so that they admit no divisor greater than unity. This furnishes us with another property. The numerator and denominator of each converging fraction are prime to-each other. qd. 324. By writing y, + for yi in the 2 Pr expression for — , art.[319 ], we obtained qi a the expression for But supposing that we had written 1 y+ ; Yo Ys -- &e. as far as the expression went, for ¥1, the same expression, we should evidently have obtained, upon reduction, the frac- tion which the continued fraction repre- sents. Similarly, if in the expression for Pn , we put for yn nm ] Yn + Ynpi+t &e.’ continuing the expression as far as it goes, we shall again obtain the original fraction. Representing then 1 Yor Ynpi-p &e. by Y, and the original fraction by F, and writing Y for 7, in the expression for a in art. [319], we have F = Pn-1 bd TE Pa-2 Qn -1 a “p Qn—-2 To / ARITHMETIC AND ALGEBRA. ‘This expression will lead us to the several 115 original fraction by its several converg- errors we commit by representing the ing fractions. For ee os Pn-2 1 Bp-1 iv + Pn-2 x5 Pn-2 Qn-2 Qn-1 aK -|- Gn-2 a f{Pn-1 Qn-2 — Qn-1 p"~*} ? Gn-2 dinate 1). (Ge Y. + Qn») Qn-2 ( ) And by a similar reduction we should obtain F % Pn-1 oe —{Po-1 Qn-2 het Gn—1 Pn—2} nine (2). Qn-1 dn-1 {qa-1 ¥ + qn-2} From these expressions we learn that the several converging fractions are alter- nately too great and too small; for if Pn-2 =l > ° F - Bard x negative, n-2 Qn-1 and vice versa. is positive, F — This result we for- merly arrived at from the way in which the successive approximations were de- rived from each other. 325. Now, in order to estimate the degree of accuracy of the several con- verging fractions, we observe in the first place that art. [320], Pn-1 Qn-2 = Qn-1 Pn-2 — ai 1. Hence, from equation (2) in the last article, ees 1 . Qn-1 Pn-1 (Qn-1 ¥ -|- Gn-2) : now Y is greater than unity, and, there- fore, Pn-1 . 1 F - is less than ———_———————__.,, Gn-1 Qn-1 (G@nu1 —- Qn-2) neglecting the sign, and consequently, @ fortior?, Pn-1 1 F - ————=. Qn-1 ts ee The following property then is manifest. The error committed by representing a fraction by any of tts convergents ts less than unity divided by the square of the denominator of that convergent. is less than Thus the error committed by repre- senting the length of the year, art. [316], 1 —— days, 7 by 365 5, days, was less than Gop or less than the saith part of aday. And similarly the error introduced by repre- senting the value of x in the equation 8 ] 2 — was less than —. 2 3 by = w ae A+Ba+Cat-+ &e. Once more, observing equations (1) and (2), and considering that Y is greater than unity, and g,-; greater than @n-», (for dna1 = Qa-2Yn-1 + Qn-g» art. [319]) it will be manifest that F — dak: n—-1l Pn-2 n= the following property. Hach converging fraction approaches nearer to the value of the continued fraction than any of those which precede it. It is for this reason that they are called converging fractions, or convergents, is less than F — , from which we derive Of the expansion of a¥ and the forma- tion of Logarithmic Tables. 326. We have already explained the purposes of logarithmic tables, art. [234], &e., and also the method of using them. We now proceed to show how they may be formed. It appeared from the nature of logarithms, art. [237], that compara- tively few of them could be expressed in whole numbers, or terminating deci- mals, and we observed that this was not necessary, since they were sufficiently exact for all common purposes when carried to seven decimal digits. This naturally leads us to endeavour to express them in the shape of a series, and if we can arrive at one which is ra- pidly convergent, neglecting those terms which have no influence on the first seven decimal places, we shall form a table accurate to the degree required, art. [292]. A preliminary step in the arrival at this series is the expansion of a’. 327. Before, however, we proceed to the direct consideration of the subjects placed at the head of this section, we must prove the following theorem, If the equation ray =atbae+tca®-+ &e. , I 116 be true whatever value be given to w and A, B, C, &c. a, b,c, &e. being in- dependent of x, we propose to prove that it is a necessary consequence of the above equation, that the coefficients of like powers of x in both sides of the equation are equal, that is, that A = a, B = 6, C =c, &c. Forsince this equa- tion is true, whatever value be given to wz, it is true when 2 = 0, and the two expressions are reduced to their first terms, so that we have Am. But A and a are independent of the value of x, and therefore being equal for one value of x, they are equal for all. We have then, striking out A and a from the two sides of the original equa- tion, the following equation subsisting for all values of 2, Ba+C2#’+&e = ba+ca?t &e. (_+y)" =l+tnytn- 5 : +n nm—l ARITHMETIC AND ALGEBRA. Divide by w and the following equation is true for all values of x. B+Cat+&e =b+Cax- &e. Hence for the same reason that A was equal to a, we have B= 0, and so on for all the other coefficients. This theorem is one of the strongest instruments in analytical reasoning, and we shall find it in the subsequent part of Algebra of frequent and extensive application. 328. We shall now consider the ex- pansion of a*, and endeavour to express it by a series ascending by integral powers of x We have already seen, art. [281], that any power of a binomial for instance (1 + y)” may be represented by a series ascending by integral powers of y. We have in fact proved that m-Iln—2 (A). This series goes on for ever when 7 is fractional or negative, but when it is a positive integer, contains only 2-+ 1 terms. Now here the remark made in art. [287] is of importance to show that even if 2 be a positive integer, we introduce no error in supposing the series to go on to infinity. The method in which we shall alter the arrangement of the factors of the several terms in series (A) will show the necessity of retaining all the terms if we wish to arrive at a general result. Returning to equation (A), and representing the exponent by 2, and writing a for 1+ y, so that y =a — 1, we have at=1+a4(a- )+ax ~ ¢ We here observe, in the first place, that only integral powers of a enter into the expression, and in the next, that all the terms after the first have x for a factor, so that 1 is the first term of the result arranged according to the ascending powers of x. In order to find the second term, or, which comes to the same thing, the coefficient of w, we proceed as fol- lows. Observing the several terms of the series, we see that in the second the coefficient of # is (a—1), in the third, 1 = 5 (a — 1)’, inthe fourth, — 7 x -; ] (a — 1)? or + 5 (a — 1)3, and so on for — sy 1 — ere pone" C0 a Are, 2 x- (r—-1) (B). (a -1)" + &e. the remainder of the coefficients ; or that in each term it is the product of the second terms of the numerators of all the factors (except w, which has no second term) which compose the coefficient, divided by the product of the denomi- nators of all the factors. Now the second term of the numerator of each factor is always the same as the denomi- nator of the preceding factor. A little consideration then will show, attention being paid to the sign, which is positive when there are an even number of fac- tors besides x, and negative when there are an odd number, that the coefficient of xv is (a-1) -3(a- 1°44 (a= 19- 5 (a= t+ Be, ARITHMETIC AND ALGEBRA. this series going on to infinity. It is doubtless possible by a similar method to examine the coefficients of x”, 2, &c., but the operation would be tedious and almost impracticable, and would be different for each coefficient. The theorem demonstrated at the beginning of this section will enable us to obtain _ these several coefficients with great faci- tity, and has the advantage of furnishing them all by the same process. 329. Resuming equation (B), and writing A for : 1 (a-—1)- 5 ~ 1)?+ = (a —- 133 - &e. and calling the several coefficients of 2”, 117 x3, &e. (the expressions for which we are about to investigate) M, N, &c. we have a=1tAaxe+Mz2? +N x -+ &e. yaatih a -This being true for all values of v, and A, M,N, &c. being independent of a, we may write 2 for av, and have the following equation, in which the values of A, M, N, &c. are the same as before, @=1+Az+M2 any eee ace fn) Subtracting equation (2) term by term from equation (1), and combining the terms with like coefficients, we have ae -a=A(x-2)TM(e- 2y tN (x3 - 2) + &e. Now every term of the right-hand side of this equation is divisible by x — 2, for every term has a multiplier of the form 2” - 2", and a — 2" being divided by x — 2, gives ar), tor? et ars. 2 &e. + 277? The above equation then may be written, a quotient in which there are 7 terms. oe) & 6) #66 6) 9 C7 Gye ee © making @ — z a common factor of the several ferms, at — a =(x# —- 2) {A+M (2antz+N@+o# et2)t&c}... » (4). Again, gee =a (a * — I). But by equation (1), putting v - 2 for 2, a= =1+A(e- z+M jw - zp+tN (we - 2p+ &e. Whence it appears, making x — 2a factor of the sum of all the terms which it multiplies, that at * b= 2 | so that a® -~ = a? (3 i 1) 1 Se jA+M@ - Sop Suagneum crt ee =r: NACH: a? — a? in equation (4), and dividing by f the resulting equation, we obtain Whence substituting this expression for a — 2, which is a factor on both sides o A+M( - 2) \ + N(e - 2)? &e)” (0) ) see) A+Mc@-+z)+N (2+ x 2+ 2*)-+ &e. = a’ (A+ M (@ - 2ZtN@- z)?-+ &e.) Now this equation subsisting for all values of 2 and 2, we may suppose < equal to x, and we obtain, since all the terms after A on the right-hand side of the equation vanish, A+M .22--N 302+ &e. =a" xX A; and it is evident from equation (3), and the remark which follows if, that the coefficients on the left-hand side of this equation would goon following the same law. Putting then for a’ its value from equation (1), and multiplying each term by A, we have At2Mae+3Na*-+&e. = A+ A?x+AM 2?-+ &e. Hence by the theorem proved at the beginning of this section, the coefficients of like powers of a must be equal; so that 2M = A, 3N=A xM, &e. &e From the first we obtain 2 Mey 2 from the second ne Ds ger 8 118 AS ~ 3.3 and the form of the two expressions shows that we should have for the ARITHMETIC AND ALGEBRA. 4 coefficient of x4, 5 s qe and so on for the other coefficients. Hence substitu- ting the values of the several coefficients in equation (1), A? a? A’, 28 Orie Sethi pote Woe. ae eee aderdersty CESS the series continuing to follow the same law. This is the expansion required. 330. We now propose to deduce from this expression a series for the logarithm ofanumber to any base. See art. [234]. The process, though at first view circui- tous and indirect, requires only to be understood to appear simple. In equation (C) any value may be givento a; and A, therefore, which, art. [326], is equal to 1 1 (a—1) —5@ _ Ven w a 1) - &e.,, admits a corresponding variety of values. Let us suppose A equal to unity, and represent by e the value of a correspond- ing to this value of A. We have, by equation (C), A being equal to 1, anda to e, xu? x e=1ltaet eS + ina.g 1 &: (D), and making & equal to 1, 1 1 Gta datiLakirts tipeate in ee Of this series we may take any number of terms. The summation of them is sufficiently easy, since each term, after the second, is derived from the preceding one by dividing successively by 2, 3, &e. The student can perform the operation, and neglecting the terms after the eleventh, which have no influence on the seven first places of decimals, he will find e = 2.7182818. This quantity then is known. The dis- covery of it does not at. present appear to have brought us nearer our object, but we shall find it a necessary instru- ment in arriving atit. It isthe base ofa system of logarithms called the Napier- ean, from Napier, a celebrated mathe- matician of the seventeenth century, who invented logarithms and calculated them to this base. Logarithms calcu- 1 ’ Nap. log.21 =” -—1 — : (n = ae We have thus obtained a series for the logarithm of a number in the Napierean system. We thus proceed to find it in any other, Lin — 198 Ree Bere at, lated to this base are also sometimes, though without much propriety, called hyperbolic logarithms, from their enter- ing into some of the properties of the hyperbola. Resuming equation (C), suppose x equal to unity, the equation becomes Avg cee a= Lona pet &e. But making w equal to A in equation (D) the series for e4 is the same as the We thus have (1). As we shall consider a as the base of the system for which we are forming our tables, we will put equation (1) under another form, and write for a, and represent above series for a. a=eA Ce ee ee (n-1)—5 (2 — De s(n —- 1)8- &e.,, which is what A becomes, by p. This being done, equation (1) becomes (2). From the definition of logarithms, this equation shows us that p is the logarithm of x to base e, or Nap. log. 2 = p. But =e? p=n-1-— 5 DES (n= 1 &e, Substituting, then, eee (3). 331, Taking the logarithms of both sides of equation (2) in the system re- quired, for instance, in that whose base is a, we have ARITHMETIC AND ALGEBRA. log. 2= log. e? = p log. e, or 119 log. m = log. e \@-D- 5 (a — rts (mn = 1)3 — ke.) acta CE) ] 1 since p=(n—1)- rz (n - 1)?+ a (7 —1)3 — &e. We here appear to be no further advanced, since log. e being taken to base a, we have at present no means of finding it. Equation (1) willhelp us out of this difficulty. The equation (1) of the last article a = e* shows, taking the logarithm of both sides to base a, that 1 = log. e© = A log. e, therefore lye log. e ol a the logarithm being taken in the system ‘ | ‘ whose base is a, and oe is known, since log. 2 = The series within brackets in the above equation is the Napierean loga- rithm of 2 given in equation (3). The quantity ae by which the Napierean logarithm is multiplied, so as to produce 1 log. N=M \@=0 ano (nm — (332.) In order that this equation, which affords the complete algebraical solution of the question, may be practi- cally adequate to the computation of lo- garithms, we must alter its form, for if m be any number greater than 2, it. is evident that the terms of the series in- stead of converging, become continually greater and greater, In fact, although A is equal to a-1 ->a- + —1)3 - &e. But this, it is evident, from equation (3), is the logarithm of a in the Napierean system. Representing, then, for the sake of distinction, logarithms taken in the Napierean system by log.’, and those taken to base a by log., we have, putting for A its value, in the above equation logla pits and substituting for log.e in equa- tion (4), 1 1 1 \ RPS os. Wr ean Ca ee BS baa (Hime BNO : bay " 1 5 (mn — 1)?+ : (2 = 1) Ae the logarithm of the same number in the system calculated to basea, is called the modulus of that system, and is evi- dently the same for all values of 1, de- pending only on the base of the system. It is usually represented by M; repre- senting it so, the equation becomes 1)?-+- x -1)- fe.) eee (Oo): we stated in the last article, that A could be found, being equal to (a-—1)-— = (a —_ It + (a-1)-&e., yet we should find this expression of little use in computing its numerical value. We proceed then to alter the form of the series in the above equation, and to the numerical computation of M. Putting 2 = 1--! in equation (5), and therefore x — 1 =n’, it becomes log. 1+ n/) = M " _ 3 nz +5 ij — &e.}, Again putting 2 = 1 - in equation (5), and therefore» - 1 = ~ n/, it becomes ; log. (1 - n/)= M4 - nl — pie - =n ~ se. Subtracting this equation from the last, and observing that log. (1-+- 7’) = log. (1 — 2’) = log. 1+ n! ——_——_ Bee ois 120 powers become doubled, we have 1+ n ih = log. ARITHMETIC AND ALGEBRA. _. Ki and that the series within the brackets.go on fol! the even powers of m/ destroy each other, and al owing the same law, so that all 1 the terms involving the odd td ; 7/3 '5 "= 2M \n ++ +e}, To reduce this to a more favourable form, let 1 +-.n! 1 — 7’ the solution of which simple equation gives List = 7 Substituting for log. Fem {27 € 4 5 (Ga) + b+e 3 (333.) The series here is convergent, and rapidly so where the difference be. tween 6 and c is very small. It is con- venient to make 4 and ¢ consecutive numbers, so that 6 — ¢ is equal to unity, and we can by means of the above Series, observing that log. b = log. — + log. ¢, find the logarithms of all numbers suc- cessively. Supposing, for a moment, that we have found the value of M for Brigg’s system when the base is 10. It being equal to 43429448, so that 2 M is equal to 86858896, we have the following rule which is taken substantially from Dr. Hutton’s mathematical tables. Call s the sum of any number (b) whose logarithm as sought, and the number ( c) next less by unity: Divide .86858896 by s, and reserve the quotient ; divide the reserved quotient by the square of s, and reserve this quotient; divide this last quotient by the square of s, and again reserve this quotient, and thus procecd continually dividing the last quotient by the square of s,as long as division canbe made. Then write these quotients under one another, the first uppermost, and divide them respectively by the uneven numbers 1, 3,5, §c. Add all these last quotients together, then the sum will be the loga- ' b ; rithm of 3? os given by equation (6). And therefore to this logarithm, adding also the logarithm of c, the next less number, the sum will be the required logarithm of b, the number required. & ce / n ; , and 7! in the above equation Lb =. = a ee a ee ee ee ee) to bt bo W 9 nt wt tt 9 o & a Oo ©& KR ee eB a ee oe a ee bt to bw bw tO © cw bd tt It =e me | ~~ = SS De ee ee ee ee bp bw & b& ‘é wwwwow wow ww w a 8 t bt = G ) ROS eo aa candi tothe 0nd btka naan een caomames eal oantein sane esoe ron mNG. omen seMcEaeNaND-—cmarecmeetiiete Tt fm yc Tie a> ee ao BK) ee poh NY = 8S bo tb bw WS wow wo w Se ee ae oe oe fap SK me A me JM oe EX =e | ee to tbe bw & wb www Ww bo - OS Oo So & a oe ee tb wp bw bw b& © to to bw b& wOwww wo re wOW WwW Ww _ ke PP KK HK eS KEE YP Ye to b> to b 0 bt bt & W bo wowwow wo wow ww w _ Pk Pk PP a a a a ee et a et ho b> 1 bO w © bt tb wowww ow kkk Pe PR eee ee to & 2 & 29 ee ee me FH WO HO ee er ae ee oF oT Or OT Ct Oo Gt OW OO OT a ee ee ee ees ee fo w ts wp wo mw w Anam on oo or or on oe Sd Co ol cole od m Hm GC GC Ow WO & 0 ww w AAAAM TABLE OF ANTILOGARITHMS. PROPORTIONAL PARTS. : 0 1 Oe Se tee 5 6 7 8 | 9 ||1}2)3/4/5|6/7|8)9 — ee eo) A Le ee EE les all ead ean ee Seed ee eo 50 || 3162 | 3170 3177 | 3184 3192 #3199 | 3206 | 3214 | sear | s2a8 |}1)1|2]}3] 4) 4] 5| 6] 7p 51 || 3236 | 3243 | 3251 ) 3258 | 3266 H 3273 | 3281 | 3289 | 3296 | 3304 |] 1) 2) 2) 3) 4) 5] 5) 6) 7 52 || 3311 | 3319 | 3327 | 3334 | 3342 ff 3350 | 3357 | 3365 | 3373 | 3381 |} 1] 2) 2/3) 4) 5/ 5| 6 74 53 || 3388 | 3396 | a4o4 | 3412 | 3400 f 348 | sia6 | 3443 | 3451 | 3459 |} 1] 2) 2}3) 4) 5) | 6) 7] 51 || 3467 | 3475 | 3493 | 3491 | 3499 | 3508 | 3516 | 3524 | 3532 | 3540 |] 1] 2) 2) 3) 4] 5) 6) 6) 7 55 || 3548 | 3556 | 3565 | 3573 | 3581.9 3599 | 3597 | 3606 | 3614 | 3622 || 1) 2) 2) 3] 4) 5) 6) 7) 7 56 || 3631 | 3639 | 3648 | 3686 | 3664 f 3673 | 3681 | 3690 | 3698 | 3707 || 1} 2) 3) 3] 4) 5) 6) 7) 8 57 || 715 | 3724 | 3733 | 3741 | 3750 3758 | 3767 | 3776 | 37e4 | 3793 || 1) 2) 3) 3) 4) 5) 6) 7) 8 5g || ssoo | as11 | 3919 | 3ge8 | 3937 | 3846 | 3855 | 3864 | 3973 | 3882 || 1) 2) 3) 4) 4) 5] 6) 7) 8 59 || 3890 | 3899 | 3908 | 3017 | 3926 ff 3936 | 3945 | 3954 | 3963 | 3972 || 1] 2| 3] 4) 5] 5) 6) 7) 8 60 || 3981 | 3990 ; 3999 | 4009 | 4018 4027 | 4036 | 4046 | 4055 | 4064 |] 1] 2) 3) 4) 5) 6) 6| 7) 8 61 || 4074 | 4083 | 4093 | 4102 | 4111 ff 4121 | 4130 | 4140 | 4150 | 4159 |) 1 | 2) 3) 4] 5) 6) 7) 8} 9 62 || 4169 | 4178 | 4198 | 4198 | 4207 § 4217 | 4207 | 4936 | 4o4G | 4256 || 1] 2 | 3/4] 5) 6} 7/ 8) 9 63 || 4266 | 4276 | 4985 | 4295 | 4305 f 4315 | 4325 | 4335 | 4345 | 4355 |) 1) 2) 3) 4) 5) 6 7] 8} 9F 64 - 4375 | 4385 | 4395 | 4406 ff 4416 | 4426 | 4436 | 4446 | 4457 |] 1] 2) 3] 4) 5) 6) 7) 8 9} 65 || 4467 | 4477 | 4497 | 4498 | 4508 #4519 | 4529 | 4539 | 4550 | 4560 L/ 2) 3)4] 5) Gh 7) 8p oF 66 || 4571 | 4581 | 4592 | 4603 | 4613 f 4624 | 4634 | 4645 | 4656 | 4667 || 1} 2/3] 4) 5) 6) 7) 9/10) 67 || 4677 | 4688 | 4609 | 4710 | 4721 ff 4732 | 4742 | 4753 | 4764 | 4775 || 1] 2) 3) 4) 5) 7) 8) 9) 10 : 6s || 4736 | 4797 | 4808 | asi9 | 4931 f 4a42 | 4853 | 4964 | 4975 | 4987 || 1) 2/38} 4]) 6} 7) 8) 9 10F 69 || 49g | 4909 | 4990 | 492 | 4943 4955 | 4966 | 4977 | 4989 | 5000 || 1] 2) 3)5) 6) 7) 8) 9 10} 70 |} 5012 | 5023 | 5035 | 5047 | 5058 #5070 | 5032 | 5093 | 5105 | 5117 |} 2) 2) 4) 5) 6) 7) 8) IU : m || 5199 | 5140 | 5152 | 5164 | 5176 fl 5188 | 5200 | 5212 | S224 | 5936 |} 1] 2) 4) 5 | 6) 7) 8) 10) 11) ne || 5248 | 5260 | 5272 | 5284 | 5297 | 5309 | 5321 | 5333 | 5346 | 5358 |{ 1} 2} 4) 5} 6) 7} 9/10) 11 "3 || 5370 | 5383 | 5395 | 5408 | 5420 f 5433 | 5445 | 5458 | 5470 | 5483 |] 1] 3) 4/5] 6) 8) 9) 10) 11 74 || 5495 | 5508 | 5521 | 5534 | 5546 ff 5559 | 5572 | 5585 | 5598 | 5610 |} 1] 3/4) 5) 6) 8) 9) 10) 12 75 || 5623 | 5636 | 5649 | se62 | 5675 f 5689 | 5702 | 5715 | 5728 | S741 |] 1} 3) 4) 5) 7) 8) 9) 10/12 "6 || 5754 | 5768 | 5781 | 5794 | 5808 # 5821 | 5834 | 5848 | 5861 | 5875 RESELLER APS iin ye 77 || seas | 5902 | 5916 | 5929 | 5943 | 5957 | 5970 | 5984 | 5998 | Gol2 || 1] 3) 4) 5 | 7) 8) 10) 11) 129 vg |! 6026 | 6029 | 6053 | 6067 | 6081 f 6095 | Gloo | 6124 | 6138 | 6152 1] 1} 3] 4) 6) 7} 8/10) 11/13) 79 || 6166 | 6180 | 6194 | 6209 | 6223 | 6237 | 6252 | 6266 | Gasl | 6295 |} 1} 3) 4) 6 | 7) 9) 10) UI 134 80 || 6310 | 6324 | 6339 | 6353 | 6368 | 6383 | 6397 | 6412 | 6497 | 6442 |] 1 | 3) 4/6 | 7) 9/10) 12) 18 a1 || e457 | 6471 | 6486 | 6501 | 6516 f 6531 | 6546 | 6561 | 6577 | 6592 || 2} 3) 5) 6) 8) 911) be 14} g2 || 6607 | 6622 | 6637 | 6653 | 6663 f 66s3 | 6699 | 6714 | 6730 | 6745 |] 2] 3) 5) 6) 8) 9) ll) 14 83 ||.6761 | 6776 | 6792 | 680g | 6323 | 6830 | 6355 | 6871 | 6897 | 6902 |} 2] 3} 5) 6 | 8) 9) 11) 18 144 g4 |! 691s | 6934 | 6950 | 6966 | 6932-8 6998 | 7015 | 7031 | 7047 | 7063 |} 2} 3) 5) 6 | 8} 10) 11/18 15) 33 || 7079 | 7096 | 7112 | 7199 | 7145 @ 7161 | 7178 | 7194 | 7211 | 7228 || 2] 3] 5) 7) 8) 10) 12) 18 154 36 || 7244 | 7261 | 7278 | 7295 | 7311 @ 7328 | 7345 | 7862 | 7879 | 7396 || 2) 3} 5| 7 | 8) 10) 12) 13 158 37 || 7413 | 7430 | 7447 | 7464 | 7499 f 7499 | 7516 | 7534 | 7551 | 7563 |] 2) 3 | 5) 7 | 9/10) 12) 14) 164 gg || 7586 | 7603 | 7621 | 7638 | 7636 § 7674 | 7691 | 7709 | 7727 | 7745 || 2] 4) 5] 7 | 9) UL) 12) 14 16 9 || 762 | 7790 | 7798 | 7816 | 7834 § 7852 | 7970 | 7889 | 7907 | 7925 || 2) 4) 5] 7 | 9) 11) 13) M4 16 § 3 90 || 7943 | 7962 | 7980 | 7998 | 8017 § 035 | 8054 | 8072 | 8091 | 8110 }/ 2) 4] 6) 7) 9) 11) 15) 15) 1% y ‘o1 || sigg | 8147 | 8166 | 8195 | g204 f saa2 | 8241 | 8260 | 9279 | 8299 |] 2) 4] 6) 8 | 9/11) 13) 15 175 ge || gaig | 8397 | 8356 | 9375 | 9395 § 8414 | 8435 | 8453 | 8472 | 8492 |} 2 | 4 | 6 | 8 | 10/12) 14) 15 175 93 || 8511 | 9531 | 8551 | 9570 | 8590°f s610 | 8630 | 8650 | 8670 | 8690 |} 2 | 4 | 6 | 8 | 10) 12) 14) 16 13} oa || s7io | 8730 | 8750 | 8770 | 8790 @ sgio | 8831 | 8851 | 8872 | 8892 |} 2 | 4) 6 | 8 | 10) 12/14 16 | 18 § 95 || 8913 | 8933 | 8954 | g974 | 8995 f 9016 | 9036 | 9057 | 9078 | 9099 |] 2 | 4 | 6 | 8 | 10) 12) 15 | 17) 195 96 || 9120 | 9141 | 9162 | 9183 | 9204 § 9296 | 9247 | 9268 | 9290 | 9311 |} 2] 4} 6 | 8 | 14) 13) 15) 17 19] -o7 | 9333 | 9354 | 9376 | 9397 | o419 ft o441 | 9462 | 9434 | 9506 | 9528 }] 2] 4] 7] 9 | 1) 13) 15) 17 205 98 || 9550 | 9572 | 9304 | 9616 | 9638 | 9661 | 9693 | 9705 | 9727 | 9750 |} 2] 4] 7 | 9 | 1) 13) 16) 18 20§ 99 || 9772 | 9795 | 9317 | 9340 | 9863 f 9886 | 9903 | 9931 | 9954 | 9977 2)/5/7)9);l1 Hha6 18 | 204 ERRATA IN SOME OF THE EDITIONS. Page 86, column I, line 38, for (v-} a)’ read (x-+-a)® Page 87, column 2, line 27, for w read x” Page 88, column 2, line 23 of note, for a read « line 24 ditto, for (1+ 2)"+* read (1+ 2)"* line 27 ditto, for (1+ 2)"t* read (1+ 2)™™ The same in lines 32 and 35 Page 90, gael 1, line 2, for (1+ x)? read (1 — x)? Lt je 2 ‘Tu line 6, for (1+ @%) read (1+ x) line 19, dele — m at the end of the line 1 ml line 20, at beginning, for ie read — m. "a m ae n 2 column 2, line 5, for . read 3 : mi(m—1 m— 1 line 7 from bottom, for Aa Ee as read | eee ar ah m(m—1).. -(m—p +1) 1) Lp isk he bo ; y2 1 Page 91, column ], line 26, for read & ae ae line 27, for fea y read VJi+e ne 27, sy jas Vv a az Page 92, column 1, lines 4 and 5 should stand thus: ] 1 an | oe 1 ln (1+ 27) ST a, Eee Pe bd ] 1 t csertl ches PRL + —.——— * ——_—... a8 + &e n 2 3 hi : 1 line 8 thus: WNeatipi fyi 2 ¥ 7 a” nN 2 / a2” 1 1 Bo eS en Ait tbarebeaa | Acc, Cowe’t-oee Oat, Beg Maree } ine ALS - ,) 3 “am + Cc. | ty i gay . line 12, for + ——— read + —.— ; l1n—1 2n—1 V 3) Ai : —.—-. fy BR line 13 thus: | ar Gs Fea Tar ae ie &e. 5 column 2, line 40, for f read $ 2 az Jave 93, column ‘ for ~ Page column 2, line 5 of note Pog cs Gta line 9 ditto, for (2: a— me read (= a— a=) ate ate@ Page 94, column 1, line 2, dele comma at the end of the line. line 5, for (a2) read (a-- a)” Page 96, column 2, line 22, for [292] read [297]. EXAMPLES OF THE PROCESSES OF ARITHMETIC AND ALGEBRA. To prevent any misconception as to the use of this treatise, we state that it is intended only for those who study the principles of arithmetic and algebra, and the reasons of the rules laid down in those sciences. The plan we should recommend is the following :—Let the student repeat examples of each rule upon paper, choosing the most simple numbers which ean be found, as well those given in this work as others, until he is capable of solving such instances mentally. Let him then proceed to the cases which contain more compli- cated numbers or expressions. This is by much the shortest way of proceed- ing, and eventually the easiest. We presume a knowledge of the four fundamental operations of arithmetic in whole numbers, and shall therefore content ourselves with showing how xamples may be formed which shall contain their own verification. As soon as the pupil knows the pro- cesses of addition and subtraction, let him take a series of numbers, each of which contains one more figure than the preceding ; say 154, 2879, 31673, 200104, and 7172618, Let him sub- tract each of these from the succeeding as follows :— 2879 31673 200104 7172618 154 2879 31673 200104 2725 28794 168431 6972514 Let him then add all his results, to- gether with the least number chosen. The result ought to be the greatest - number. 6972514 168431 28794 2725 154 7172618 As an exercise in multiplication, let twe numbers be written down for the student, each of which he is to multiply by itself. For instance, 142 and 361. | 361 xX 361 = 130321 143. 142= 20164 Subtract 110157 ' Let him then take the sum and dif- ference of the two numbers first chosen, and multiply these together, which should give, the same result as the preceding. 361 361 142 142 add 503 219 subtract. 503 X 219 = 110157 For division, let the student multiply two numbers by themselves, and divide the difference of the results by the dif- ference of the numbers; which should give their sum. But the division of any two numbers by one another may be made, and the result verified by multiplication as usual. SECTION 1.—Common Fractions. OPERATIONS containing fractions with very high numbers are of little prac- tical use; decimal ee pre- 2 _ EXAMPLES OF THE PROCESSES ferred. But as exercises of arithme- tical accuracy we shall give, among the rest, a few cases of high numbers. L—To reduce a fraction to its lowest equal to it which has a smaller nume-. rator and denominator. Principle employed.—The value of a fraction is not altered by dividing both its numerator and denominator, or mul- terms. Definition. — A fraction is in its lowest terms, when there is no fraction tiplying both its numerator and deno- minator by the same number. 1 2 3 4 a id cal c = Ni : ; ll equal. 5 ‘i ; : ¥5 &e., &c., are all eq 3 6 9 12 ys tes ty a oma aes » &c., are all equal. 7 14 21 28 rected ia hae, 2 Rule. Divide both numerator and denominator by the greatest whole num- ber which will divide them both without remainder. Case 1. Where it is evident that a certain number will divide both numerator and denominator without remainder, and that the result is in its lowest terms. | Heat 1 8 18,02: 27 9 Sai ves 48°” 4 14°08 99° 11 15. 5 Point out here by what numbers the numerators and denominators are divided. Case 2. Where it is evident that the numerator and denominator are divisible __ by some number, but not evident that the result is in its lowest terms, divide by that whole number, and proceed as in Case 3. (Observe that Case 3 may be employed without this, if preferred.) A number is divisible by Two, when the last digit is divisible by ‘wo, or even; as in 66, 48, 132. Three, when the sum of its digits is divisible by three, as 162, in which 1+ 6+ 2or 9 is divisible by 3. : Four, when the two last digits are divisible by four, as in 16864, in which 64 is divisible by 4. Five, when the last digit is either 0 or 5, as in 180, 965. tiply by 2, and strike off the cipher.) Six, when it is even and divisible by three, as 486. Seven, according to no rule sufficiently simple to be useful. j : Eight, when the three last digits are divisible by eight, as 2794216, in which 216 is divisible by 8. Nine, when the sum of its digits is divisible by nine, as 729, in which 7 + 2 oly or 18 is divisible by 9. Ten, when the last digit 1s a cipher. : Eleven, when the two sets of sums made by taking alternate digits are either equal, or differ by a multiple * of 11, as 1034,in which 1 + 3 is the same as 0 + 4, 121 in which 1+ 1is the same as 2, 129382 in which 1+ 9 + 8 or 18, differs from 2 + 3 + 2 or 7, by 11. Twelve, when it is divisible by four and three. : The preceding rules may be applied to the following fractions : find out which is employed in each, (To divide by 5, mul- bes asada 0a, B7a0 = 1144 104 in the lowest terms. 7944 1986 662 © 8916 7 2299 > 743 (Case 3.) 8904 1272) 212 4494 = 642 ~ 107 (Case 3.) Case 3. When there is no very evi- less, the divisor by the remainder, the dent divisor of the numerator and. last-mentioned remainder by the new denominator, divide the greater by the remainder, &c., &c., (as afterwards | ® A multiple of gis any number which can be divided by a without remainder. OF ARITHMETIC shown,) until there is no remainder, or until it is evident that two successive remainders have no common divisor. AND ALGEBRA. 3 therefore 58 is the greatest divisor which 4466 and 1856 have in common. Tn the first case, the last divisor used 58)4466(77 58)1856(32 will divide both terms of the given ANB Ll fraction, and will reduce it toits lowest 406 116 terms ; in the second case, the fraction 406 116 1s already in its lowest terms. oF iO *»* Observe that whatever divides 4466-77 two numbers divides their difference: Therefore —— = — therefore 102 and 107 can have no com- 18560" 32 mon divisor; if they had, it would be bee atin either 5 or would divide 5. This will Reduce 8209 to its lowest terms. _ often be useful. 1847)8209(4 4466 4 7388 Reduce 1856 to its lowest terms. {sea 821)1847(2 1856)4466(2 1642 3712 205)821(4 754)1856(2 820 1508 1)205(205 348)754(2 205 696 Kou 58)348(6 348 0 This tells us the greatest common divisor of 1847 and 8209 is 1, or that there is no divisor which will reduce the fraction to lower terms. 2433 3 156933 329 Leea7 TT 19bAe ay 314175 355 100110 355 100005. «113 31866 113 7992 «9 54369 63 11544" «13 73355 85 instances of higher numbers, 7241379310344827586206896551 63 9999999999999999999999999999 97 42614574994432 _ 16807 149720237927424 59049 _ The following is a table containing some prime numbers (or numbers which have no whole divisors greater than 1) by which examples may be formed, 23 367 857 1637 3299 8443 18583 29 397 883 1709 3389 8573 20611 83 433 947 1759 4591 8669 32801 149 509 953 1831 4673 9011 43717 179 541 967 1847 5189 915] 58573 181 619 971 1861 5407 918] 60013 19] 647 977 2081 6329 9403 72053 257 709 983 2111 6449 9521 84229 271 761 be 2287 7237 9631 97073 311 809 S92 2749 7321 9967 99991 Take any two of the preceding numbers, say 23 and 149; multiply both by any number, say 8, giving 184 and 1192, then Bb J reduced to lowest terms, gives i 1192 149 Many thousands of examples may be thus formed, N 4 EXAMPLES OF THE PROCESSES Il.—To reduce Fractions to acommon Denominator,— That is, to find fractions having the same denominator which shall be respectively equal to a set of fractions having different denominators. Case 1. When the fractions have denominators, of which all the divisors can be easily seen. An example of this case will be better than any rule. To reduce to a common denominator the following fractions, 1 2 1 3 5 1 1 2 3 1 13 ree my ee Soe ae tome ae a write down all the denominators which are not evidently divisors of some of the rest. 7, 9, 10, 12, 16 Write these down in prime factors, that is, make them by multiplication. “f 3X3 5 xX 2 2x2x3 QR 2K Mee Take each prime number as often as it occurs in that one of the preceding which has it most often. 07 PBS b'7 Multiply all these together, which gives Q9x2x2x2x3x3x5x7 = 5040; Divide this by all the denominators in succession. 5040 — 2 = 2520 5040 — 7 = 720 5040 — 3 = 1680 5040 + 8 = 630 5040 — 4 = 1260 5040 —- 9 = 560 5040 — 5 = 1080 5040 — 10 = 504 5040 —6 = 840 5040 — 12 = 420 5040 - 16 = 315 These need not all be formed by multiply every numerator by the result actual division, for it is clear that to of its denominator in the preceding list, divide by 9, we may take the third part and we shall thus have the numerators of 1680, in which 5040 has been already of the fractions required, while 5040 divided by 3. int will be the common denominator, as Now look to the original fractions: follows :— , 2520 1 xX 2520 = 2520 is 1s ——— 2 6040 2. 336 2 xX 1680 = 3360 — 1s si 3. 5040 1 xX 1260 = 1260 : is wait i 4 6040 Similarly 3. 3240 ue La 0 —is WHS & 5040 9 5040 5. 4200 Ss 1542 6 5040 10” 5040 120 tat 7S 5040 12° 5040 a" 2680; 13. 4095 8°” 5040 16” 5040 Fractions given, The same reduced toacommon denominator. Bias 7 295 250 42 8 12 100 600 600 600 Se ei eae m9 20 7586 12 16 30 240 240 240 240 aan ay Iemma eae DS a IDS , 3 * Consider this as 7" OF ARITHMETIC AND ALGEBRA. 5 The most convenient common deno- minator is the /east number which is divisible by all the denominators or their Zeast common multiple; but any common multiple will answer. The Jeast common multiple is found in the preceding process. Case 2.— Where the least common multiple of all the denominators is evi- dent. This, generally speaking, is when the denominators are very low num- Numbers given. 2, 3, 4, 6, 49) 10; 12%. 18: 2, 6, 10, 6, 8, 10, Soe Sau 9 5.) 8, 40: 15: ue a a as Ge 2, 4, 6, 8, 10, 18, 20, 24, 16, 18, 22 bers, and the least common multiple is found by multiplying the denominators together, rejecting any factor out of each, which is evidently contained in a preceding one. For instance, the least common multiple of 4 and 6 is not 4 xX 6 but 4 x 3, because the factor 2, which is thrown out when 6 is made 3, is already in 4. The following are instances :— Least common multiple, 2 5 exo we EIS Ait, Sn: De Sk ee Se Le oar yee 2 5G) ae Xe RE Se SO & Xa xX 5 SF E20 ot Bix Bl aaa Su X02 So XU Sh S26 7 RO KIM PS ges) 1966 Di KN a Oe we BO bea Oe 18 X10 >x) 2% =) 360 lO Xe 9s C= ISse When the least common multiple has been found, perception derived from practice, rather than rules, must be the guide; if that fails, go back to Case 1. Fractions given. 1 2 1 5 Gamer ered 3 3 1 al ae 9 5 1 Sai Gers 7 3 i) ish aolkeg Tee Gare eee ee d. sages Case 3—When there are only two fractions with complicated denomina- tors, either multiply numerator and denominator of each by the denomi- ~ nator of the other; or, if considered worth while, find the greatest common measure of the two denominators, and their quotients when divided by it; multiply each numerator and deno- minator by the quotient of the other denominator. 3 11 To reduce ei and — to a com- 82 25 Fractions given. 53 27 181 936 113 355 355 113 Reduced to a common denominator. 6 8 3 10 FOr i bly 9 Lopdy ve 12 6 7 48+) gee og 81 15 i] had akg tae birt 1s 42 Yi 50 90 90 90 30 20 15 1D 10 CO. LAOS GON 460° ° 1 60 mon denominator. 33 _ 33 x 25 | 825 82 82x 25 2050 ll Ll-x<. 82 902 95. 25 x 82 2050 81 To reduce Gao and en to a com- mon denominator. Here the greatest common measure of 1540 and 7700 is 1540; therefore the fractions are 81 185 7700 7700 Reduced to a common denominator. 49608 4887 169416 169416 12769 126025 40115 A0115 be ANS 0k ER ER Tis a RS Oe TR * For 24 write 2, because 6 is already a factor of 18, and of the residuary factor 4, 2 is already in 20, . a vee , of bers, % ie 6 EXAMPLES OF THE PROCESSES IlI.— Estimation of the Value of Fractions. Rule 1. When fractions have a com- mon denominator, the greater has the greater numerator. 16 9 14 13 12 12 li 11 Rule 2. When fractions have a com- mon numerator, the greater has the less denominator. 16 16 91 91 F< a8 Gb Aas Rule 3. If the humerators of two fractions be added for a numerator, and the denominators for a denominator, the resulting fraction lies between the two first. 1 ee ] % lies between 5 and 4 S45 3 5 7 lies between To and 7 23 20 — jj en — and— - lies between ao i Rule 4. By adding the same number to both numerator and denominator of a fraction, the fraction is brought nearer to 1; by subtracting the same number from the numerator and deno- minator, the fraction is removed far- ther from 1; that is, addition to both decreases fractions greater than 1, and increases fractions less than 1; sub- traction from both increases* fractions greater than 1, and decreases fractions less than 1. is a continually increasing series. Si 4 Wee gs 28a eneien is a continually decreasing series. te a 10 7 is less than =, y 4 - a 27 99 3S greater than a8 1V.— To add and subtract Fractions. Rule. Reduce the fractions to a com- mon denominator; do with the nume- rators what is directed to be done with the fractions; let the result be the nu- merator; let the common denominator be the denominator. Reduce the result to its lowest terms, if thought worth while. he Ad 1 : What Is 5 + id These are . and 2 1 | * hence 5 ei 3 has 3 + 2 for numerator, pad 5: Pent and 6 for denominator, or 1s ° Simi- says Dah Ae Wee aia & oud ne) a 6 5b Ole wig BT 8 es 3 Ty > 3339 7 38 Ri TOE A AE Peds se Sha BS A 6 SB i Bar erg i373 > O71 (EON ee Hone | Se TOU) ihe 30 A NS Hanan lige +9 Bey a 53 cd 19 20082 89 847 30883 153 18329 ~ 4281 os ans) ci OF ARITHMETIC AND ALGEBRA. 7 LO eee Otek: 1 1 23 by the denominator, and add it to the 2° 3 + Aivt BG AO numerator; let the denominator re- main LEG Gn Oe. ore 1g PO and 6. 6G, Har URE 1 64 Ben S445. tah ae a Mi RS 4 [ayy 87! 60 oe wilt ee! Such reductions as the following are p ? 1) one particular cases which often occur : ; yok. 16035 3 93 298 elev idie & Ve 100-100 100 100 sii heh Mle Paice Aa 11 361 pant ees Rule. Multiply the whole number Si 795 07,9, UF V.—To multiply or divide Fractions by a Whole Number. Rule. Do as directed with the nu- 2 merator, or the contrary with the de- either 35 nominator ; that is, to multiply, mul- 6 vt Lae hk tiply the numerator, or divide the the former is the more simple. denominator ; to divide, divide the nu- merator, or multiply the denominator. ce x ts 15 3 Multiply 2: by 7, 8 80 Myles Gerd 6 Bh hae bal either a5 OF 35-7 that 1S, 144 988 pnt} aoe 107 * > tor either — or+ Soin As the latter is the more simple. ee @ Divide 3 either — a5 or 35x 3° that is, Multiplication of Numerator Division of Denominator Division of Numerator Multiplication of Denominator 4 Multiply 7 by 14, or 7 X 2, ‘has 7 16 7 75 Pee . 19 + = 96° 144 . 12 — to7 ~~ 1? = Fo7 When the multiplier is composed of factors, it may happen that some fac- by 3 tors may be most conveniently used in 5 ; one way, some in the other. The stu- dent must render himself very familiar with the following: \ is Multiplication, \ is Division. pony 60 Divide W by 25 or 5 X 5; AX 24:8 60-5 12 777% Lik 72a 5 385. 4 16 fy. et ap * B= 3 o1 147 12 _ 24 2. ua 39° 29 33 39 = 78 [D8 2. oe 1 4Be eee meee 35 x 20 = a 35°" 70 55 , = 205 55 Ae 6a. oe 63 /° 126 25 2 —~ X 86 = 50 8 ‘EXAMPLES OF THE PROCESSES VI.— To multiply and divide Fractions by one another. Definition 1. The product of : and 4 This is the answer to such questions as the following: What is two-thirds of =? What is four-fifths of 3 A. gave B. ; of his share, and B. gave C. = of what he got. How much of A.’s share did C. get? If1 gallon cost 5 of a shilling, how much of a shilling does : of a gal- lon cost? What is : taken two-thirds of a time? What is twice the third part of =? &e. Definition 2. The quotient of = = divided by 4 ‘This is the answer to such questions as the following : What number of times, or what parts of a time, 2 4 2 does 3 contain a How must = = be treated, so as to give 3 =; that is, into how many parts must : be divided, and how many of these parts must be taken, so 2 ey that ° may result? If 1 gallon cost 3 of a shilling, how many gallons, or how 4 much of a gallon, may be bought for = 5 of a shilling ? Rule. To multiply, multiply a es by numerators, and denominators by denominators. To divide, divide numerator by numerator, and denominator by denominator; or invert the divisor, and multiply. Or to perform either operation, BN ee multiplier or Sey and proceed as in the other. Multiply 3 aye a divide the product by °, and multiply the result by Ze wu a OD a 8 8. 5 8 79:96 56 7. £392 g%5° 3x5 18 IST ISS 7B) 75S 11! OBS 16 20 320 4518} 20 Serre a Tonk 2016 a20 Bd ase gt io ea Wath ett Before the multiplication is made, strike out any factors which are common to a numerator and a denominator ; before the dzviston is made, strike out any factors which are common to both numerators, or to both denominators. 16 = 35 ope a4 Li Bi Bie AAD 9 Ta) a me te Pg Boas) Any eS. BS) AK ae SIM KS ee 55 28 oak Bele KH. Te oat Leah Giallo ae Lived: Sowa e eee get ah. 390 5 EB ad aes ee oe Al Led. Bae oop go Be od Sib mag Peo a pe eT 21 10 _ 15 FS 27 Sue (88 16 ‘f 8 18 10 4 Se bg 51 51 dl 132651 Be SGM uy 2 aD 9 ga ae 64 96 * 96 * 95 ~ 884736 ag 3 30 ges yee OF ARITHMETIC AND ALGEBRA. 9 113 | 221 _ 24973 Al2 ,, 397 _ 163564 355. 118 ~ +# 41890 167 25, 277. °° 129759 Aime ssn! «238 93°. 126 _ 11625 169.9% 963 162747 4600 662 30452 60. G0 5 60 1 G0) 60 Y 60 _ 46656000000 61 Gh 2C6h rem Gh pa Gl 51520374361 Weg, SEN Rape a pera py a A Te Aa or ae pe tom Geli PS weg oN 38 975 Leased 3 5 12 tea 1s 25 cele: 635 11S. 338 + 13447 Ley 25 324 pete Vo 9126 3163 . 799 _ 79075 ps igh at 468 ° 25 373932 rae Uae Re LY ae hele hae ee ky i Glee, LATO LP Seo 135 7x Ex 8) + 6B oo Dace tit Vi ate pillhe pba: Tok 112 ay ak fans bo ts Ley 20 ik gee ee: Ey 7 2 Cee el a oe ries pane 9 ae: ane: 8] 8 3 8 pp lees af eles ee 3 10 Goaee eta a? 6 LG le 14 2 4 ES ON 100 . 101 _ 100 ae 5 7 3 5 101 9 GY xsi eS Orcas te id Gt i Oy ba aes cy eas ie 9= 12x tax = ViU—Fraciions having Fractions in the Numerator or Denominator, or both. Rule. To reduce such fractions to equivalent simple fractions, multiply the numerator and denominator by the least common multiple of the denomi- nators of the fractions contained in them. 1 39 To reduce ea the a simple fraction. 2 en 3 1 3 : 3- * 6 a O 8 +3 _ 21 1 1 a 2 ae MS — * ‘6 : 3 ; 3 ete i To reduce : . to a simple fraction. at won Dae i 7 10 —- 2 ie pleiyN| = That is, 4 7 EXAMPLES OF THE PROCESSES | The least common multiple of 7,4, 2, and 7, is 28: 1 —_ Ko28 = 22 1 6 2 x 28 =" 2 Se 8 lll 1 f ‘ diminished by 2 3 which 53 is of 122. 1 oo | — Be pete | (0.2) Wile yn © pole} dole py] co aa 9 13 a a me Wy Pe ee os — 5 i 104 2 117 a 8 Se Brera dh: Se Sued Dyas Pe 4 3 5 eg ye: 8 4 Hr may ha . : zs when diminished by 27 is the same proportion of 6 3 2 1 11 oh 168 you 9 fe ty 10 1749547 1000 ~ 4200000 1 5 Braet 6 a p48 3 a BL 2. [2 ;) ate 14 rf ge Ce 1 3 2 Wiles a 3 eee | 9 4a 3 Rees eh air waa eee a nS 1 BL 0'3 he es : te i * ; 7 ; i) Salat tue cia ih, tal WA pa Ng Se eee 3 1 eae (1 Daas WE 17 81 11 9 Mage Gee ah 2 hee to 3830/1 Manes Bee 16h) 815 NB 13 8. eon Mee en | io Liebe Ben ee _ 2 ene OF ARITHMETIC AND ALGEBRA. NEL ge yh ek 1 ‘ ae: ik © ee, 2 mM oe 2 ae Y is 7 5} 5} crx ety et -s 4 a 4 Tata ts aie sere 4 Fit term 4 4 4 Rp wre Seer tat og gee Sa 8 7 en ae Pe et eae 1X.— Verification of Algebraical Processes. The following are some algebraical equations which are always true, what- ever numbers or fractions may be placed instead of the letters; provided only, that wherever a subtraction occurs, such as a — Jb, a must be _ greater than 6. The student must at- tempt to verify them; and the proof that he is correct consists in his finding the same number on each side of the equation. For instance, in the first example, let a stand for 4, and b for 3: then 1 ote b 3 De, Eo Rat, ee Ea | Fg ey LU ma ah SEAMS Oe Dla ; ar 3 2 a- 2 T 9 Gir bn aon ake (7) Again, ‘yl 1 aa Me 2 pay ab+ bb Fic ead ee OA The following list may be considered long, but it must be remembered that every one is also an example in algebra, and that the young student cannot* be more usefully employed at this stage of his progress, whether the operation be considered with reference to arith- metic or algebra. The student should first try each expression with some whole numbers, before he proceeds to use fractions, in order to be certain that he understands the meaning of the terms. DO, & ale pee G45 5." 6. Ob 418) a-b ,a+6b _, 2aa + 2a6' Bob! Wie bl aaron | 1 1 Ae a TG 4e3a:) \ oe aah dae (a + 6) x (a ~ 6) = aa — bb (a + b) xX (a + 6) = aa + 2ab + Ob (a — b) X (a — 6) = aa — 2ab + bb AL el Ah th apc Wire A ie A hg oon Sg hangs x (v-+a)x(ex+b) = xx+(a+b)xx+ab (w—a)X(x@—b) = xuv-(at+b)xXx+ab (ax — by) (aw — by) = (aa + bb) (we + yy) — (ay + ba) (ay + 6x) OHO a Po G 2 2 BE 2 Fe Wo Di 1 xx —- 1 r+i1 xxx + 9re~ + 2674+ 24 _ xvvex + 672% + lle + 6 rata n tl +1) M@t+% _ 9 ad Ce i Ace ai ah 2 we—-3rt+-2 “4-2 weo-1loe+9 «2-9 exet+4 x+ il MED OED +3) _— 2M+EVM+ D~MM+V iy yyy 6 6 2 * The very little power which even advanced students generally possess, of turning their algebraical into arithmetical results, is : l and is a very serious impediment to higher studies. is one of the principal features of school instruction, as it exists at present, 12 EXAMPLES OF THE PROCESSES OF are TOP ae Oe bbbe eae eas aaa (a — b) Cat Cy HOO ne SP bbbe Gob Se ‘aa aaa aaa (a + b) 1+a2 1—2 32 - 1_@+D@4)_, gl, @tDG-1 fee PR Gee Eee Pak a (@tb+o(b+c—aj(ect+a—b)@+5—o)= = 2aabb + 2aacc + 2bbce — aaaa — bbbb — cccc a -b X.—Algebraical Theorems of Approximation, for Verification. If p be very nearly equal to 1, then the following theorems are nearly true : BPS Raye Peceeaee ak yee ek ppp = 3p — 2 at et ia eats 2 coe he pppp = 4p — 3 &e. 1— ppp = 4 — 3p &e. If a be very small, the following theorems are nearly true. 1—32 1+ 6a 6 — 4@ 8 NC) ge an ees Tea nek 3. ete Fee If w be very great, the following theorems are nearly true : BS Hea as Here ies 32 Von ee x+i1 xL r+ x Lehi eS 2 re Section 2.—Decimal Fractions. 1.—Eercises on the Meaning of the Decimal Notation. *1 is read decimal, one. *123 is read decimal, one, two, three. 36°012 is read thirty-six, decimal, nought, one, two. The student should now write the following and similar tables :— 1 1 *"l me —: . — : a ans i0 01 means i00 001 means 1000 “9 iv, gn oun) Aiea “hoo aN (kes 10 100 1000 3 3 3 25 e re "03 ee @ ern = °003 @'.8, @ ar Res 10 100 1000 “4 2 “04 i “004 ; o e se 10 e . } e 100 e es e 1000 &c., &e. Ning 2 3 100 20 3 123 *193 Is —— — ——: whi i ae —— : which is —— 10 * Too” 1000? WMP 18 Th59-* ipoe * Goo? WES on9 ae 4 100 4 104- *0104"isi--— ———; which is —— + ——-: which is ——— 100“. To000? “Uh 1S Gop * T0000: 10000 ; 7 SA tetera’ a!) 7 pie Dp iy, 6°7 is 6 + Tp Which is +5 + To? which is 10 OF ARITHMETIC AND ALGEBRA. 13 35°0138 = 35 + : aa 2 oe a 100 1000 ~ 1000. Ay biedh Pahl 8 2008 }) ans 1000 — 1000 eth 1 7 4 174 Sat 00 + 7000 + T0000 — 10000 12°11 12 : ~ a BAEe es 10 100. ~=100 12345 = 10000 + 2000 + 300 + 40 + 5 5 1234°5 = 1000+ 200+ 30+ 4+ : faa CULE UME TR SY RF ee 10 . 100 ae 4 5 12°345 = 1 ee ee Ly 4 0+ 2+: 79+ Too + i000 2 3 4 5 | ee Me 1 — — arises pis chek a9 uy 100 + 1000 5 10000 12345 = 1234°5x 10= #£123°45 x 100 = 12°345 x 1000 1234°5 = 123°45x10= #£12°345 x 100= 1°2345 x 1000 123°45 = 12°345 x 10 = °#1°2345 x 100 = -12345 x 1000 172345 = *12345 x 10 = +012345 x 100 = -0012345 x 1000 °12345 = ‘012345 x 10 = *0012345 x 100 = °00012345 x 1000 1 9 45 . c . ) ° a toatl 345 _12°345 _ 123°45 _ 1234°5 10 100 1000 10000 yep 12°345 123745 _ 1234°5 _ 12345 10 100 1000 10000 12 *AA 9, . ° pits 123°45 _ 1 34°5 12345 a 123450 10 100 1000 10000 eA ne 234°5 _ 12345 _ 123450 _ 1234500 10 100 1000 10000 8°2 = 8°20 = 8°200 = 8°2000 = 8*20000, &e. Instances like the preceding should be continued until the student is SO familiar with the changes of the decimal point as instantly to point out the effect produced by it, without recurring to a rule. Il.—To find a Decimal Fraction which shall be nearly equal to a given Common Fraction. Principle. No common fraction has a decimal fraction exactly equal to it, unless its denominator is divisible by nothing but 2 or 5, or is composed of the product of some numbers of twos and fives. But a decimal fraction can be found, which shall be as near to a given common fraction as we please, though not exactly equal to it. Rule. Annex ciphers to the nume- yator, divide by the denominator, and neglect the remainder. Cut off as many places from the quotient as there were ciphers annexed to the numerator, for decimals. If one cipher was annexed, ] the decimal so obtained is within 10 of the given fraction; if two ciphers, MA £ Vit ris sy dp within Too? if three ciphers, within 1 T0002 and so on. In this and all other decimal opera- tions, when directions are given to cut off a certain number of places, and there 14° are not places enough to be so cut off, affix ciphers to the beginning, in suf- Jicient number to make up the defi- ciency. Thus, to cut off three decimal places from 25, write 025; to cut off ten decimal places from 118, write °0000000118. Find a decimal fraction which shall 1 F 18 10000” 23° Annex four ciphers to 18, and divide by 23. be within 23)180000(7826 rem. 2, Cut off four places from 7826, and the ‘answer, *7826, is within EXAMPLES OF THE PROCESSES Find a fraction which shall be within 1 ] 7000000 * 933° 9 13)1000000(1095 rem. 265. Make siz decimal places in 1095, which gives *001095, the fraction re- quired, Definition. A decimal is said to be true to the (first, second, third, &c.) place of figures when any alteration in the (first, second, third, &c.) place of figures would remove it farther from the truth than it is as it stands. For instance: 99 — ©1616 very nearly. 10000 It is also very nearly -6161, but not 18 quite so near to this as to °6162. The of 23° second is a little too great, the first a Tnale rape : little too small; but the second is not . cae ——_“ _—- so much in excess as the first is in de- VR MERLION Ge. Tang at 230000’ fect. | 1 pay ' Rule. To make a decimal true tothe 230000 115000? Which is less last fieure, find one more figure than 2 Anas Mi 450000" 115000 is wanted ; if the last figure be 5, or in 1 upwards, increase the preceding by 1. n 10000° Thus: * 18829976. If we wish to retain one place only, write *2 a ay two places - - °19 ” a9 three ay Rat - pe 188 0 99 QUE ie sei yh BBB %» %» five - = = = °18830 9 %9 SIX sis oe bl 188800 4s Ap seven - = = = °1882998, wha is the nearest decimal fraction mt any ah — 1°73883 5 to B00 true to five places of decimals. er sh 4h ewi0'0669), 10) ALO dy aes Annex six ciphers to 1, and divide by 741 555 oh 16 355 309)1000000(3236 “ae = *0554 vs = S' 14759 Answer: ° 003236, which, made true to five places, is *00324, WEA omy Aaa Ly eesonage The following examples of decimal 596 3090 fractions are all true to the last place : 700 100 ame peel OTT aoe ce RARBG 793 633 24 873 AK a ET I EOI a ES Lae x = *3929 S = 07376 * The student must not be surprised at this fraction haying the same decimal (to four places) 1 447 as the preceding. ‘The two fractions do not differ 24 on) *0345 Pees = 1°73930 — by so muchas one ten-thousandth, OF ARITHMETIC AND ALGEBRA. Instances of the above process carried to a: greater number of places : 653 ere 1°0315955766192733017377567 140600315955, &e. 43 : 3530 *04512067156348373557187827911 85729275979619 1 Bers *00178253119429959001782531194295900, &e. Tnstances of fractions which can be exactly expressed decimally ; that is, in which the denominators are made by multiplying éwos or fives, or both: Binah og Li +9 t 9 dive 044 2 5 4 25 qe tan E0605. eee 08086 8 i oh eg sgos ie Lk. = 008% 4 ue ee 0078 EAE 64 125 128 1 8 +901953125 9. = * 013671875 512 512 63 _ .994375 101 _ + 9193991015625. } 8192 I1l.— Reduction of Decimal Fractions io a common Denominator. 15 Rule. Annex ciphers to all which have a less number of places than are in that which has the greatest number same number of places.+ Thus °*1, “12, *123, reduced to a common denomina- tor, are °100, *120,.°123. of places, so that all shall have the Fractions given. ‘031, 2°4, Reduced to a common Denominator. °0600, °0310, 0148 12°300, 2°400, “1G?, °0148 3197 °06, , 12°38, 1V.— Addition and Subtraction of Decimal Fractions. Rule. Proceed in every respect as in 12 From 66°112 whole numbers, but keep decimal points 12°] Take 2°01783 under one another, and place the deci- 1°42 64° 09417 mal point of the result under the other "0081 points. (See page 1 for methods.) 05°5281 Add *12,' 12°, 1°42, and’. 0081. 1 “1 + 0) + 2001 + “0001 = I°litl 1 2+ -03 + °004 + °0005 = 1°2345 67 + 7°8 + °89 — 1°2168 = 74°4732 6°718909 — 2°1488 = 4°570109. V.—Multiplication of Decimals. Throw away the decimal points, and mal places in the result as there are in all preliminary ciphers; multiply the both multiplier and multiplicand. results together, and take as many deci- Multiply together the following : 32 6°3 2°99 “001 6°0 Multiplicands. ¥en) “84 “O11 01 -5 Multipliers. 12 63 299 1 tr 6p 11 84 11 1 ie 121 252 3289 1 300 504 5299 1°21 5°292 "03289 00001 3. Answers. 16 EXAMPLES OF THE PROCESSES BSS cs 64 SS Nei Rise G a4 80 xX +8 644.008 *'*08 = 15°94 X °004716 X RODS 7 Buse og °00064 800 X °0008 = 254°0836 = 4050°092584 *22240656 = ‘00104886933696 08 % 98 == 1064 °64 °923521 X% °28629151 = *26439622160671 °155 X 24°025 = 3°-723875 14°2 K *142 = 2°0164. VI.—Division of Decimals. Rule. Case 1. When the divisor has no decimals, or is a whole number, pro- ceed as in common division, and let the Jirst decimal place of the quotient be that figure, in the making of which the Jirst decimal place of the dividend is brought down; but if more than one 9)173°43 19°27 Case 2. When the divisor has deci- mal places, strike out the decimal point, and remove the point in the dividend as many places to the right as the number of places which have been thus destroyed in the divisor, previously an- “09)1°68(. *4)°0192( 9)168°00... 4) *192 16°22... “048 2°5)°1793 25)1°793(7172 76 43 25 180 175 50 50 0 "07172 18)°0041(° 9002 36 decimal place of the dividend is used in making the first figure of the quo- tient, put the decimal point first, and then a cipher for every decimal place after the first which is used in making the first quotient-figure. nexing ciphers to the right of the divi- dend, if necessary. In both cases, ciphers may be an- nexed at pleasure to the right of the dividend, and used in forming addi- tional quotient-figures. "1193 11)300°00... 27°27... Quotients. *0025)179°3 25)1793000(71720 175 43 25 ~ 180 71720 Quotient. Case 3. If the dividend be a number followed by ciphers, as 86400, strike out the ciphers, proceed as before, and when the process is finished, remove the decimal point one place to the left for every cipher so struck out. 1°793 2500)1°793( es a hy Le 25)1°793(' 07172 te °0007172 ) 2500-e ee ee Al 1 A d F Q pools SL Col in which it bole ae t It irst Quo- | Altered Divi- mn in which it must Dividend. | Divisor. Dividend, ob nt tidat igure: peer eerie xa stand. ; 4 gives it. 1° 9628 64°19 | 196:28 6419 3 196°28 2nd Decimal Place. -0019 °134 1:9 134 1 1-90 2nd Decimal Place. 674 °012 674000 12 5 67 Ten Thousands Column, 6:22] *9136 62210 9136 6 62210 Units Column. *7021 | 123-65 WOE2 Lie 123869 ib 70°210 3rd Decimal Place. 1 ‘001 1000 1 1 1 Thousands Column. 118 190°5 1180 1905 6 1180°0 | Ist Decimal Place. 1 116°4 10 1164 8 10:000 | 8rd Decimal Place. OF ARITHMETIC AND ALGEBRA. 17 6 6 06 006 eon tk re etihee ‘001 rare ae 600 600 06 Tg = 1000 Tog = 10000 “G09 = 00001 8°4 8°4 Sai ous ay yin aeh a td 4 eae rece Al AOLn i ee 84 ‘ “084 “084 "0084 ioe wees terre ao 0012 = 7 1 2 T[59 = 6289308 59°79 = (00862069 Ee dan eure 6821691° 97627 wil yt pts Se 937°6567. ONC alae BS ai peat alae 24336 ERS Beale 007396 156 86 . 61000 ee c26 F800 = 000007393483 -g2g3 = 73939'393939 59 OR 30000 = * 000007375 D00s79 = 39723"66148532 When the student has acquired suf- ficient knowledge of the meaning of decimals, and expertness in using them, he will need no other rule for all the cases than the following :—Put a semi- colon in the place where the decimal point ought to be, in order that the result should contain no higher or lower denomination than wznzts, that is, should lie between 1 and 10; pass from the semi-colon to the decimal point as Divisions required. Places of the semi-colon. it stands, repeating ‘tens, hundreds, thousands, &c., as successive figures are passed over, 2/ to the left, and TenTHs, HUNDREDTHS, THOUSANDTHS, &¢., as successive figures are passed over, IF TO THE RIGHT. Let the first figure of the quotient have the denomination last named. We give underneath the place of the semi-colon, and the value of the first place of the quotient. Value of tirst places of the quotient, 84 Whig eg. 5630 84 “31 5630 : PON aes LVL OIG AIG Lieidaas uisoOe [ee Lata 369°7 216°4 369°7 216°4 49872°3 193 °2 49;872°3 19332 LOGE Nth VIl.—Contracted Multiplication of Decimals. Rule.—To multiply two decimals together, so as to retain only a certain number of places in the product, with- out the trouble of finding the rest— invert the order of the figures of the multiplier, and write them.under those of the multiplicand in such a way that what was the units figure of the mul- tiplier may come under the last place of decimals, which is to be retained. Multiply as usual, with this exception, that each figure of the multiplier begins with the figure of the multiplicand which comes immediately over it, the figure next to that being only used to carry from (as in the subsequent exam- ple). Put the several lines directly under one another, instead of removing each one place to the left. *,.* As it is almost impossible to make this rule clear in words, we sub- join an example at length. Ex. To multiply 147°3861 by °6457, retaining only three places of decimals. The second factor, written so as to show a unit's place, is 0°6457, and in reversing, the 0 must fall under the third decimal place of the other factor, thus :— C 18 EXAMPLES OF THE PROCESSES 1473861 Multiplier reversed ; units place 0 falling under third decimal 6 of the — 75460 upper line. | a Multiplier 6; figure to begin with, 8, figure to carry from, 6. Six times 88432 6 is 36, nearest ten, four tens, carry four. Six times 8 is 48, and 4 is 52, put down 2 and carry 5. The rest as usual. 5895 Multiplier 4; figure to begin with, 3; figure to carry from 8. Four times 8 is 32; nearest ten, three tens, carry three. Four times 3 is 12 and 3 is 15, &c. The rest as usual. 737 Multiplier 5; figure to begin with, 7 ; figure to carry from, 3, Five times 3 is 15; nearest* ten, ¢wo tens, carry two. Five times 7 is 35 and 2 is 37, &c. The rest as usual. 103 Multiplier 7; figure to begin with, 4; figure to carry from, 7. Seven times 7 is 49, nearest ten, five tens, carry five. Seven times 4 is 28 and —- 5 is 33, &c. The rest as usual. 95°167 Add as usual, and mark off three places ; (the number proposed) for decimals. The full product of 147°3861 and *6457 1s 95° 16720477, which in thousandths only is nearest to 95°167, our result. The following multiplications have the proper arrangement and result given. No decimal places means that the whole number of the result is required, with- out fractions. No. of Arrangement of Multiplication required. Decimals Multiplier and Result. retained. Multiplicand. 36°3771 x 9°99339 three 36°3771 : 933 999 Se eae e Lod ‘ 9 7 O° j 19°@81137 x 523°36 two 19 081137 9986°30 6 3325 °0699268 x °9975641 seven °0699268 . is 1 4657990 0697560 13763819 x *05877853 one 13763819°0 : 35877850 0 et ots 794504°4 X 7° 986355 none 753554°1 ‘ 5536897 6018150 1°2799416 x 6156615 seven 1° 2799416 % 5 1665160 #830108 multiplier has no figure above it; but the carriage from the 3 (9 x 3 = 27) is three tens, and three must be written under the right hand column of the preceding lines. Where the figures of the multiplier extend to the left of the multiplicand, continue as long as there is either mul- tiplication or carriage. Thus in the first example, the first 9 of the arranged Vill.—Contracted Division of Decimals. of the divisor are cut off. When the abridged divisor is not contained in the — Rule.—Proceed as usual, until the number of quotient figures remaining a to be found does not exceed the number of figures in the divisor. Then, in- stead of annexing a cipher, or bringing a figure down from the dividend, cut off the last figure of the divisor; that is, do not employ it except to carry from, as in the last rule. See how often this abridged divisor is contained in the remainder ; multiply, carrying from the figure cut off; finda new remainder; cut off another figure from the divisor, and repeat the process until all the figures * In this case, 15 is equally near to one ten and two tens. take the higher of the two. remainder, eut off a second figure from the divisor, put a cipher in the quotient, and proceed, We subjoin a detailed example. To divide ‘1299494 by °9915206, as far as nine decimal places. The first quotient figure being a decimal, and there being seven places in the divisor, two quotient figures must be found by the usual method ; after which, the process is explained. It is usual, and generally moze correct, to OF ARITHMETIC AND ALGEBRA. -Divisor, afterwards 1% abridged. Dividend. Quotient. 9915206 )12994940( *131060717 9915206 30797340 29745618 991520 6 1051722 : : Cut off the 6, reserving it to carry from; 991520 is contained in 1051722 once; 991521 once 6 is G6, nearest ten, one ten, carry one. The rest as usual. 99152 06 60201 Figure to carry from, 0; 99152 not con- 9915 206 tained in 60201, cut off another figure from divisor, and put 0 in quotient. Fi- gure to carry from, 2; 9915 contained in 60201, six times. Six times 2 is 12; ; nearest ten, one ten, carry 1. Six times $949] 5 is 30, and 1 is 31. The rest as usual. 991 5206 710 ‘ 991 not contained in 710; cut off one more 99 15206 figure, and put 0 in quotient. Carrying figure | ; 99 contained in 710, seven times. Seven times 1 is 7, nearest ten, one ten, carry one. Seven times 9 is 63, and 1 is 694 64. The rest as usual. 9 915206 16 Carrying figure 9. Divis. 9, contained once in 16.- Once 9)is 93 nearest ten, one 10 ten, carry one. Once 9 is 9, and 1 is 10. 9915206 6 i J No divisor, carrying figure 9. What num. ber of times 9 will carry 6, or be most 6 nearly 60? Seven times 9 is 63; put 7 “ane in the quotient, and carry 6, which 0 finishes the process, No. of Decimals Dividend. Divisor. to be retained. Quotient, 1 3° 14159265 7 *3183099 ] 2°7182818 7 *4342944 2992°9 id Oe iv al Y 2 5 57° 80347 171°8 414°487636 Y) *414487636 °273 74° 529 9 * 003663004 *9008202 *67272804 8 * 00121922 When the divisor itself contains more places than are required in the quotient, as many places may be cut from the right as will make the two the same; and the dividend may be cut down in the same way until no more places are left than will give one figure in the quotient, to the abridged divisor, re- membering the rule for increasing the last figure. Thus °1648267 — -7263, to two places only, may be found from °16483 — 73, by the rule exemplified above. Both in multiplication and division, it is best to retain one more place than is absolutely required to be correct. Section 3.—EH2traction of the Square Root. Hxamples of Surds and Irrational . Quantities. l.—Eviraction of the Square Root. The rule for this will be better un- derstood by a detailed example than by any verbal explanation. Though the quantities operated upon are deci- mal, it is to be understood that a whole number may be used in the same way. For 5, for instance, is 5° 0000, &e. The following contains the working of the rule at length for the extraction of the square root of 32°19, to four places of decimals. Annex so many ciphers that the decimal point shall be followed by twice as many places (eight) as there are to be decimals in the root (four). This gives 32°19000000. Point the unit’s place, and every other place from it, to the right and left, which give 3219000000. C2 20 EXAMPLES OF THE PROCESSES Given number Root found, figure Divisors. pointed. by figure, as below. 3219000000 (5°6736 First period 325 nearest square, 25; root 5. Put 5 in the root, 25 and subtract 25 from 32. 106) 719 Remainder 7; bring down next period, 19. Double 5 (10), which place in divisor. Cut off one figure from 719,—71. This contains the divisor 10 seven times ; try 7, as follows: annex it to divisor, 107 ; multiply by it, 107 X 7 is 749: thisis greater than 719: 7 will not do. Try* 6. Then 106 X 6, is 636—less than 719. Put6 in the root — and in the divisor, and subtract 636 from 719; remainder, 83. Bring down next period, 00; add 6 last found to 106, giving new divisor, 112. Cut one figure from 8300—830. This contains 112 seven times. Try 7, and 1127 X 7 is 7889. Put 7 in the root and in the divisor, and subtract 7889 from 8300. Remainder, 411. Bring down next period, 00; add 7 last found to 1127, giving new divisor 1134. Cut one figure from 41100, —4110. . This contains 1134 three times ; trial * no longer ne- cessary. Put 3 in the root, annex 3 to divisor, giving 11343. Subtract 11343 X* 3, or 34029. Remainder, 7071. Bring down last period, 00; add 3 last found to 11343, giving new divisor 11346. Cut one figure from 707100 ;—70710. This contains 11346 six times. Put six in the root; annex 6 to divisor, giving 113466. Subtract 113466 X 6, or 680796. Remainder, 26304; less than half of 113466, which shows that there is no occasion to change the last found 6 into 7, to have the nearest decimal of four places. 636 1127) 8300 7889 11343) 41100 34029 113466) 707100 680796 26304 The required root is therefore 5°6736 ; by which we mean, that though 32:19 has no exact square root, yet 5°6736, multiplied by itself, will give a result xearer to 32°19 than any other number with four decimal places. This we will try. Multiply the three suc- cessive fractions, 5°6735, 5°6736, 5°6737, each by itself, retaining five decimal places in the product. 5°67350 5°67360 5° 67370 53765 63765 737/65 2836750 2836800 2836850 340410 340416 340422 39715 39715 39716 1702 1702 1702 284 340 397 32°18861 32°18973 32°19087 Find the difference between each of these, and the quantity which we first set out with, and we have °00139 00027 *0008, of which the second is the smallest. down a second period, and place a cipher in the root and the divisor. The following is an instance in the extraction of the square root of 100406552374249. Where the calcu- lator would simply annex a cipher or When, after cutting off one figure from the altered remainder, the divisor is not contained in the result, bring period to a line, we write the line again with the cipher, that the student may see the several steps. _ * This trial will rarely be necessary after the second step. So that having cut one figure from the increased remainder, the number of times which the divisor is therein contained may be written down — ou the right, and the whole divisor, thus altered, multiplied by its last figure, OF ARITHMETIC AND ALGEBRA. 21 100406552374249(1 1 2) 00 20) 0040 10 i‘ 2002) 004065 1002 4004 2004) 6152 200403 615237 100203 601209 200406) 1402842 20040607) 140284249 10020307 140284249 a eee Wherever a dotted line occurs, the augmented remainder, with the last figure cut off, is found not to contain the dividend, anew period is brought down below the line, a cipher is an- nexed to the divisor and to the root (also brought down), and the figure e e e e e e which, after this, answers the purpose, appears at the end of the divisor and of the root. There being no remainder at last, the exact square root require is 10020307. : The student should perform the pre- ceding operation in this form : 100406552374249(10020307 1 2002) 004065 4004 200403) 615237 601209 20040607) 140284249 140284249 0 We must notice one more case in which a cipher may occur. We will first write the beginner’s attempt, as it would be if he were not cautious. To extract the square root of 2034: First Attempt. 203400, &e. (45° 1 16 85) 434 425 901) 900 901 In the first he has gone wrong, for though 900, stripped of its last figure, contains 90 once exactly, yet 901 (the new figure being annexed) is not con- tained in 900. He therefore puts a Corrected Process. 203400, &c. (45°09, &e. 16 85) 434 425 9009) 90000 81081 9018) 892900 &e. cipher in the divisor and the root, and brings down another period. The de- cimal point of the root always precedes that root figure in forming which the first decimal period was used, annexed —_— OO OTT ® The preliminary ciphers may be omitted} 20 is not contained in 004 or 4. 22 ciphers being always considered as de- cimals. If periods of ciphers be thrown away in the beginning of the operation, the root is all decimal, and has a ci- pher at the beginning for every period so thrown away; but this rule does not apply to the throwing away of a single cipher (not a whole period) at No. given. Dos painted a | 0*1000 &e. 85 0°8500 &e. “0683 0° 068300 &e. “0068 0006800 &e. 9°79 9°7900 &e. 97°9 979000 The preceding method may be short- ened, as soon as half the decimal places required have been found, by substi- tuting a contracted division. The rule is, when Aalf the number of (decimal and other) places have been obtained, instead of forming a new pe- 12°00 . 9 64) 3 00 2 56 686) 4400 4116 6924) 28400 27696 69281) 70400 69281 6928201) 11190000 6928201 692820|2) 4261799 4156921 104878 69282 35596 34641 955 693 262 208 54 EXAMPLES OF THE PROCESSES Loewe aT the beginning, or to a cipher in the ' unit’s place. In the following examples, the num- ber whose root is to be extracted is in the first column; the pointing at full length in the second; the same with the decimal point and preliminary ci- phers, if any, thrown away, in the third; and the answer in the last. Do., simplified. sane 1000 &e. 31622776602 8500 &e. “92195444573 | 68300 &e. 261342686907 6800 &e. 082462112512 97900-&c. 312889756943 979000 Se. 9°89444988 riod, let the remainder stand, strike off a figure from the divisor, and proceed as in contracted division. The following is the extraction of the square root of 12 to 12 decimal places by this method: ~ 6 « « (3°464101615138 55 The nearest. The 8 not so much too great as 7 would be too small, OF ARITHMETIC AND ALGEBRA. The student may furnish himself with - examples to any amount by the follow- ing principle: If A has the square- root B, four times A has the square- root twice B, nine times A has the square-root three times B, and so on. Let him then choose a number or frac- tion, and extract the square-root, say of four times that number, as well as of the number itself. His first result 23 should be twice the second. The last figures only cannot be expected to agree. The extraction of the cube-root is a long and useless process. When the student becomes acquainted with lo- garithms, he will always use them for the extraction of all roots, the square- root included. Il.—Dejinition and Notation of Powers and Roots. Operation. Denoted by Lest 7? eo Te Ow: i Vite Ge Ae ie Gat | 7: Tid. A ee Me eae 4° &e. &e. Commonly called The square orsecond power of 7. The cube, or third power of 7. The fourth power of 7. The fifth power of 7. &e. By analogy, 7 is written 7! and called the first power of 7. Cancion fat | Manner of denoting p. filled by p. pp=—7 MW DE TEs.” eo bs ppp =7 “7 or 73 pppp =7 ‘V7 or7é . ppp = 7% ‘7 or 73 ppppp = 7§ W6 or 78. : « pp = 71 Vimor7%. . Verify the following equations by multiplication : 16 = 24 = 4% = 83 = = 38= 9) = 273 = 817 =243 32= 8% =42 256= In such an equation as 32 = has two names ; one referring it t 4, the other to the 32. 5 is called the exponent of 4. § is called the logarithm of 32 to the base 4. ib bo tals {| (=>) ms cath 169.0 to wn Name of p. Square, or second root of 7, or 7 to the power of one-half. Cube, or third root of 7, or 7 to the power of one-third. Fourth root of 7, or 7 to the power of one-fourth. Cube root of the square of 7, or 7 to the power of two-thirds. Fifth root of the sixth power of 7, or 7 to the power of six-fifths. Square root of the eleventh power of 7, or7 to the power of eleven halves. Verify the following assertions : The numberlis the logarithm of the undermen- | corresponding num-| to the base tioned ber undermentioned 2, 3,4 100, 1000, 10000 | 10 3, 4,4, 4 (262144, 8,4,2 | 64 6,5,4 64, 32, 16 2 3, 45 4, 2 (1024, 32, 16, 256 $3. 3. OF 3,3 32768, 4096 rides 4, $2, 7% (65536, 4, 64 Joy 25s 4516384, 262144, 2 Ill.—Particular cases of Propositions which are proved by Algebraic Reasoning. The student should go through the whole of these, adding others similar to them if necessary, until he performs all such operations by habit, without rules. 105 =-10 «10° 102 X% 104 = 10% X 103 = 1000000 B= 9X W= 2X 2W= BX BWke. = 256 28 % QT = 9 129 % 19! = 19% 5 6 7 gy 84 Bh & 2°) Se, x, 513 8 8? 83 84 24 EXAMPLES OF THE PROCESSES pia (22)° = (23)4 = (24)? = (26) = 4096 (38)* = 332 (72)° = 755 (1009) = 1008! &e. Or, V2 = 4/2? = 4/28 = 4/24 = 14/95 &e, 4/338 = 4/339 12,/3612 Lp 4 S35 or (85)3 = 820 5/322 = (5/32)? = 4 <- <- ae rw" ot bard to am aa foe 20 » | | ~~ Co — bene oO S als elo = cafes CUES | 2 ewe Wyse vet% = 775.0 gz tgs — gtd = 104 109 = 104 = 10% — 104 = 975, or 5/94 + 3/92 = = 36 3/15 = 155 4 VJ 49 or (93)3 = 99 bf re 0/5 or (83)° = 850 5/323 = (4/32) = 8 a ae ((c10%))*)s 0 = ((aoot)') 62 X 63 = G2+3 = 6%, or V6 X V6 = %/60 r3J72 x 5/74 = IT” 109 109 = 104 10° 6/i—— 68 = l 3 a 618 = 7 §u 1 1 ' { 34— 33 = =i ener : Sone 313 (103) x (ott = (108)? = 102 x (53)" x (53) + (54)8 = 5 6 means 6° \ \ (2-30)! *. means (2-36 Ji Piss TES aS My dices RRO a = 83, or V8 + 3/8 = 6/8 OF ARITHMETIC AND ALGEBRA. 25 «} “O14 14 7 means gis 8 means 81000 3 3 3 4 4 4 1 1 1 o x4 =8 3 x10 = 30 6° xX 7 = 42° 1 5 2 L aus 23 nS} _ — pe. — DSS MAL =. 93, 96/4| « o3/240 Ob 8X AB Ala ot 4/2 tol i) 2 2g 7X8 =7%X 8] or 7? Kx NV82 = W562 1 =\2 1\?2 1 2 TaNt | 40h 19 6.96 — V2 ={- = _ 13 oe) Se ch sie oie (; 2) (G) x: 2 (Nh Sass aaa i ¥3 REM 3 2 6 =< 160 es Eeoanl Up x 14 = _ ( 0) (5 i) (5 ye) 7 N81 = 9 V81°X% 2 = 181 X V2 = 9/2 = 4/156 = 2 39 fe%rlers VI8 X 20 = 2 J90 3 8 2 V8 = NV4X2= N4X N2=I2N2 = N82 = 4 VD J44 = 2 V1 (160.4 4/10 RIL SS6r Rat Al: TO afO = (900: 743 = 4147 12 /12 = 1728 3/56 = V8 X 7 = WB X W7 = 237 168 = 2 3/21 4/288 = 416 X 18 = *W16 X 4/18 = 24/18 , 6144 = 4 4/24 4/6966) = 3 4/86). ..,.. 4430 4/768 7/256 = 27/2 Ae EOE aE Ebe aw Te ee Wea Pan yee ha On Wha | Lisa. 4/66 ae we NS, Soh ty i Yd Wa _ N66 Ta ACT ZL Se aI. Tie wet e Ly 5 5 J5 havea ah Tie et AD = RY Zo satan. 7 GaSe eT, ee: Che FEE) pa EM ae ore a 3 CN GL ite 2474 10) ot oe 7 N7 10 OF ARITHMETIC AND ALGEBRA. 27 (J2a+ 3h)? = 6+ N35 (NTE - WI)? = Bf - 44/150 (oV2+3N7)2 = 71 12V14 = (32 — 23)? = 30 — 12/6 Lox Wee Anis. Pi geet ue BNP) BS iy hs 9 _ 3 = 1] —--——™ 3 ray Ad = pa ae, Gal ) ao (Ay + 408) +206 3 -" 1--\~_ 55 8 950 (Ay tla! 4) = oh dae oe S10) = oem all | AD ee a eee ae G8: /70) 49 ag%° & Git i) 144 15 %° (it = Wt)? 2 1°31 + 89] (V6 V7)? = 13 = B42 Rule 2. The product of the sum, and difference of two quantities, is the difference of their squares : (a + 6b) xX (a — BD) = a? — 6? (6 + 4) x (6 — 4) = 36 — 16 bi By Factors given. Product. V5 + 3 V5 — WB 2 bi we ee a1 1 a/ ie 9 2 ig 1 2 Ta iG 350 | 1 ay q ‘3 a eee 3 6 + aft 6 3 353 /5 J 8 /3 55 /% a8 8 3 8 24 iit a/} 1 1 S543 27 2 4 Vi0+3 Vi0 - 3 1 Suction 4. Miscellaneous Questions involving the Use of Fractions, 2 i 1 2 3 1, If; ofa shilling buy j of a gallon, If yd. cost £5 how much will : of a shilling buy ? 2yds. ,, £2 A i. muy : gall. 1 yd. costs £5 Then 2s. buy ; gall. ; 13 yds. cost £~ 1s. buys gall. 2 vie pes idk gal 32 If £25 buy 35 gallons, how much 58. buys = gall. will eal Hid Anenie, ft 2. If = of a pound sterling is paid for ‘ : Ay tf 3 acres let for £1 03; how much = of a yard, how much must be paid for % will 11 acres let for? Answer, £365 3h? 28 iP A £5 be worth = of a sheep, and : of asheep be worth a of an ox, how much must be given for 100 oxen? Answer, £2000. 6. If 12 oxen be worth 29 sheep, 15 sheep worth 25 hogs, 17 hogs worth 3 loads of wheat, and 8 loads of wheat worth 13 loads of barley; how many loads of barley must be given for 20 41 23 = loads. 7. If 12 of A count for 13 of B, 6 of B for 18 of C, and 13 of C for 2 of D; how many of A count for 100 of D? Answer, 200. oxen? Answer, 8. A. is indebted a of his whole pro- perty, and loses 4 of it. He recovers as much as amounts to adding : to what he then has, and afterwards loses ; of what he has got. Can he then pay his debts? Answer, Yes; after which a5 of his original property will remain to him. 9. A. gains 3 per cent. (3 parts out of a hundred) on what he already has, and B. 7 percent. But A. gains £100 less than B., and they started with the same sums. What were those sums ? Answer, £2500 each. 10. There is a number to which 3 is added, and = of the result taken. To a of the result 15 taken. The produce is then 13. What this 5 is added, and was the number? = c, then 4, which is the exponent of a, is called the loga- rithm of c or of a’, to the base a. Thus 10°=1000, whence 3, the exponent. of 10, is called the logarithm of 1000 to the base 10. Hence it follows that 3 may be the logarithm of all sorts of numbers, according to the base chosen. Thus: D8 jana S 3 = log. 8 (base 2.) 33 = 27 3 = log. 27 (base 3.) A? = 64 3 = log. 64 (base 4.) &e. &c. &e. But as, in the practice of logarithms, no other base is used, except only 10, we shall, in this treatise, suppose no other base ; and logarithms to this base are called common logarithms, tabular logarithms, or Brige’s logarithms. And because we have nothing to do with 3 33 Rules. Arrangement of Tables in Method of taking out Logarithms, and Numbers to Lo- the method of constructing logarithms, but only with the use to be made of them when they have been found, we shall refer to works on algebra for the former part of the subject, and proceed to the latter, after we have stated in what the difficulty of finding them consists. According to the common language of algebra, if we raise the mth power of 10, and extract the mth root of the result, we have what is called the “th power of 10, or ~/ 10" aa 10” We shall now simply write down some results, not expecting the student to verily them; because, though that might possibly be done by ordinary arithmetic, yet the process would be of very great length and trouble: 108 or 10% or 10 =: 179952623150 nearly, 301 *301 N10 or 1920 or 10 = 1°9998618696 nearly. boa “30108 meen! 1 930403 or 10100000 or 10 = 2°0000000200 nearly. So that we may get a result as near to 2 as we please; that is, we may find a decimal fraction «x, which shall, as nearly as we please, satisfy the equa- tion 107 = 2. The answer is 2 =*30103 nearly. And in the same way we may find an approximate logarithm for any other number or fraction. These ap- proximate logarithms are arranged in tables, with certain modifications de- rived from the following fundamental 2X, oe) 10 Log. 2-+ Log. 5 = Log. 10 Log. 2= °3010300 Log. 5= ‘6989700 Log. 10 = 1:0000000 rules, which are proved* in works on the subject. 1. The logarithm of a product must be the sum of the logarithms of the factors. Thus, 6, 8, and 10, multiplied together, give 480; the logarithms of 6, 8, and 10, added together, give the logarithm of 480. The following in- stances may be immediately verified from any tables: 4X 7 = 28 Log. 4 + Log. 7 = Log. 28 Bogs 4 4 *6020600 Logs 7-5 8450980 Log.28= 1°4471580 2. To find the logarithm of a quotient, subtract the logarithm of the divisor from the logarithm of the dividend. Thus, 20 divided by 5 gives 4; the logarithm of 20, diminished by the logarithm of 5, is the logarithm of 4: 100 =~ 40 = 2°5 Log. 100 = 270000000 Log. 40 = 1°6020600 Log. 2°5 = 0°3979400 64—16=4 Log. 64 = 1°8061800 Log. 16 = 1°2041290 Log. 4 = 0°6020600 3. The logarithm of a power, root, or combination of power and root, which is * The reader must recollect throughout, that we here lay down rules only, not demonstrations. 34 EXAMPLES OF THE PROCESSES denoted in algebra by a fractional exponent, is found by muliiplying the loga- rithm of the number given by the exponent tions will set this in a clearer light : : Log. aa or Log. aaa or Log. Va Log. %/a Log. */a? Log. Ja” or What is the logarithm of the square root of 156? Log. 156 = 2°1931246 4 Log. 156 = 1°0965623 Ans. What is the logarithm of the fifth root of the fourth power of 2097 ? Log. 2097 = 3°3215984 4 5) 13 ; 2863936 2°6572787 Ans. 4. The logarithm of 1 is 0; that of the base (which is here 10) 2s 1; that of the square of the base (here 100) ts 2; that of the cube of the base (here or or 1000) 2s 3; and soon: or LOG a Loa c0 Or Lar LOU) fk LOG £0.22 ty) Log O00 gii = a Log. LO0s=12 Log. 100000 = 5 &c. 5. As the number increases the loga- rithm increases, and the greater the number the greater the logarithm: but the rate at which the logarithm in- creases 1s perpetually diminishing as the number increases. Thus we see that, as the number passes from 10,000 to 100,000 (through v2nety thousand units) the logarithm passes from 4 to 5, receiving no greater increase than takes place while the number passes from 1 to 10 (through ve units only). 6. In any logarithm (4°6183 for in- stance) the whole numbev (4) ts called the characteristic, and the remainder (*6183) the decimal part of the loga- rithm. 7. In any number (368°414 for in- stance) the figures which precede the decimal point (the 3, the 6, and the 8,) are called integers, and those which fol- low the point are called decimals. And Jigures, when opposed to ciphers, are called significant. Thus, in 864000, 4 is the last significant figure; in *000193, I is the first significant figure. or Log. Log. in question. The following equa- Ce. ee, OS tt as. = 3, uog. d. Swat. ae “ink SB ee. Zz Oo crs ete ae ao ae ’ a* = # Log.a. m a” ™ Log. a. n g. A fraction less than unity (5 for instance) has none but a negative lo- garithm: but that students may use logarithms who have not studied alge- bra, we affix a meaning to the term negative, for this subject only. The term multiplication is extended in arith- metic to whole numbers and fractions, so that multiplication, in its extended meaning, includes the first meaning of division: thus, to multiply by i is to divide by 10. But from the connec- tion which exists between multiplica- tion of numbers and addition of loga- rithms, and also between division of numbers and subtraction of logarithms, we cannot use. the word multiplication in an,extended sense, which includes division, and keep rules (1) and (2) at the same time,* unless we also use the word addition in an extended sense, which includes subtraction. And this is done as follows: by 1 we mean a unit, with a warning, that in all opera- tions performed upon this 1, we are to subtract where we should have added if the bar had been absent, and to add where we should have subtracted. And with this we say, that 1 being the loga- rithm of 10, 1 is the logarithm of me 2 being the logarithm of 100, 2 is the logarithm of nee the following are in- stances of the use of this sign, with the corresponding real operations :— Multiply Divide 1000 by = 1000 by 10 Log. 1000 = 3 Log. 1000 = 3 1 ye LORS ae ee 19 Log. 10 se Add 2 Subtract 2 And 2 is log. 100 + — 100, 1000-+ 10 = 100 1000 xX 0 ee i hi Pe DA SE SA A EA EN a Uh UL ure TE * The choice is, between making two rules, and using the words of one rule in a sense which will make that one include both. The latter is the more difficult at first, but the more convenient in the end. — — OF ARITHMETIC AND ALGEBRA. Divide Multiply 1 1000 by = 1000 by 100 Log. 1000 = Log. 1000 = 3 Log. 100 ie Log, 100:=2 Subtract 5 AILS And 5 is log. 100,000. 1000+ = 100, 000 1000 100=100,000 When a subtraction appears which is impossible, invert the subtraction, and place the bar over the result. The following are instances, with the cor- responding operations, the first line of each set containing logarithms, and the second the numbers and operations cor- responding :— Bias =2 1000 + 100,000 = = | a =1 100-1000 =>, Oi-as =~ 3 1 + 1000 = In all other cases, combinations of the preceding rules may be used: and it must be considered that 1 and 1 added make 2, and so on: the follow- ing instances will contain all the cases :— —— a, Pe ja on || S| be 100 ** 1000 OT ey or oF 3-4 2 or 100,000 Vie a eb “Gti | Iii om = 10 fs aoe eines | Inte tes age kc Gite 1 1 ro x 1000 — ies 10,000 4 =— 4 1 ir 35 What are the results of the following, and what are the corresponding opera- tions in the numbers to which the terms are logarithms ? What is the logarithm of *5? Log. 5 = *69897 Log. 10 = 1°00000 Subtract, Impossible, there- fore invert the subtraction, and place a bar over the whole; as follows, °30103 Whats, 20°35 9°52 Log. 20 = 1730103 Log. ‘5-= ‘30103 Add ——100000 What is 100 — *5? Log. 100 = 200000 Log. °5 = 30103 Subtract 2°30103 which is log. 200 But the necessity of using decimal places with a negative sign, can always be avoided, and the characteristic only made negative, as follows: for Ans. 10. \ Ans. 200. log °*5 or log. aa or log. 5 — log. 10 or "69897 - i write *69897 + 1 or 1°69897; in which the first figure only is to be used as a negative quantity. We re- peat the preceding instances. What is 20 X *5? Log. 20 = 1°30103 Log. °5 = 1°69897 Add 1°*0000 Ans. 10. Here the 1, which is carried after adding 1, 6, and 3, (where we have placed an asterisk instead of a cipher to mark the place) instead of increas- ing the 1, destroys it. What is 100 + °5? Log. 100 = 2°00000 Log. *5 = 1°69897 Subtract 2°30103 as before. To make any logarithm which is entirely negative, negative in the cha- racteristic only, make that characteris- tie greater by 1, and subtract the decimal part from 1. ny 386 ‘What is 40°41372? 1 — *41372 = *58628 Answer 41°58628 0-3 =1°7 1°21 = 2°79 0-1 =1°9 1°6141982 = 2°3858018 In the practice of logarithms, tt will be necessary to appear to subtract the greater from the less, which is done by subtracting in the usual way till we come to the last place, inverting the subtraction which there occurs, and placing the negative sign over the result. From 1°6936 20°414 Take 3°0177 29° 666 Ans. 2°6759 10°748 6°4:4.12'°RB)4*.6°6.4 000 79 «6 fea G+ Fis s 8 2°5 24 #1°9 5°72 0°0000 0°00000 2°1896 0°12345 3°8104 1° 87655 In the following examples the nega- tive characteristic is treated in the manner already described, namely, as to be subtracted in addition, and added in subtraction. The figure carried is always to be added, and, therefore, makes a negative characteristic less: thus 2 carried to 5 makes it 3. Add. Add. Add. 1°48 9°83 2°18 2°56 1°47 6°00 3°41 4°66 9°14 3°45 5°96 1°32 From 17616 8°413 4°17 Take 2°929 1°097 5°28 4°687 9°316 0°89 From 0°000 0°00 2°66 Take 2°147 1:42 3°44 1853. 0°58.s«<1:°22 _ As the difficulty lies entirely in the line of characteristics, we give some examples of that line only, the figure carried from the preceding line being written in Roman figures at the top. EXAMPLES OF THE PROCESSES I I II Iv Ill Add f° @ tg) iigmaren S10 °C.” see 7 ago Ful eee Aggie giivyioge 4 re Re NS PAL La O I I 1 From 2 0 3 1 Take 3 g 4 _0 Sel P ake ROM 2 To multiply a logarithm with a nega- tive characteristic by a whole number, proceed in all respects as in common multiplication, except only in subtract- ing, instead of adding the figures which are to be carried, so soon as the characteristic comes to be multiplied. 1°61" 125 OT Oe ae 4 8 8 2 2°44. 5°65 32°8 772 When the multiplier exceeds 12, and the process is not performed in one line, the better way is to omit the charac- teristic altogether, at first, and sub- tract the product arising from it after- wards, as in the following multiplica- tion of 2'136 by 15. °136 15 680 136 2°040 4 X 15 = 60 Subtract 58°040 To divide a logarithm with a negative characteristic by a whole number, begin by ¢ncreasing the characteristic until it is divisible by the whole num- ber, make the quotient a negative cha- racteristic for the result, and use the augment which was found necessary, as ifithad beena remainder. Thus, to divide 1°4 by 2, increase the first 1, and make it 2 (necessary augment, 1) and 2 being contained in 2 once, 1 is the characteristic of the quotient. Then, taking the augment 1, prefix it to the 4, giving 14, which contains 2 seven times. Therefore 1°7 is the quotient. OF ARITHMETIC AND ALGEBRA. 2)3°010 4)1°11 = -:10)6°52 2° 505 1-78 T:45 7)8°10 3)9°12 4)4°13 2°87 “3°04 1°03 As divisions by higher numbers rarely occur, we shall only give one instance, that the student may exercise himself in reconciling the process as it here appears, with the rule given. The asterisks mark where the process dif- fers from common division, 13)165°61(13°35... 13 35 * 39 TAG 39 71 We can now give a logarithm, by help of the tables, to any number or fraction, and can, by the above conven- tions, make the rules marked (1) (2) and (3) include all cases of logarithmic operations, by help of the following rules. (a) An alteration in the position of the decimal point, alters only the cha- racteristic, and not the decimal part of the logarithm, if the significant figures remain the same: thus all the follow- ing numbers and fractions have the same decimal part in their logarithms, with different characteristics. *000256 2°56 25600 ° 00256 25°6 256000 °0256 256 2560000 *256 2560 25600000 (6) In every whole number, let a decimal point be understood after the unit’s place. Thus 58 is 58°, or 58°0, or 58°00, &c. (c) When there are figures before the decimal point, let the characteristic be one less than the number of places of those figures. Thus the logarithm of 26861°5 has the characteristic 4; so also has that of 26861 (or 26861"). The decimal places of the logarithm of 21925 are *3409396; hence Log. 21925000 = 7°3409396 Log. 2192500 = 6°3409396 Log. 219250 = 5:3409396 Log. 21925 = 4'3409396 Log. 2192°5 = 3'3409396 Log. 219°25 = 2°3409396 Log. 21°925 = 1°3409396 Log. 2°1925 = 0'3409396 37 (d) When there are no figures (or only ciphers) before the decimal point, let the characteristic be negative, and let it tell in what place following the decimal point, the first significant figure is found. Thus, in * 0000136, the first significant figure being in the fifth place following the decimal point, the characteristic of the logarithm is 4. The decimal places in the logarithm of 324 being °510545, we have Log. ‘324 = 1°510545 Log. °0324 = 2°510545 Log. *00324 = 3°510545 Log. *000324 = 4°510545 Log. *0000324 = 5°510545 The decimal places in the logarithm of 1 being 000, &c., we have the follow- ing logarithms, which consist entirely of characteristics : Log. 1000 = 3°0000... Log. 100 = 2°0000.. Log. 10 = 1°0000 Log. 1 = 0°0000 Log. ‘1 = 1°0000 Log. .01 = 2°0000 Log. *001 = 3°0000 &e. and these are the only numbers to which logarithms can be exactly found ; the decimal places of all others being approximations only. Tables of logarithms (generally) con- tain the decimal part of the loga- rithm, which is evidently all that is necessary, as the characteristic can be found by the preceding rule. Being approximations, they are more or less correct according to the greater or smaller number of places which they give. Modern tables never have fewer than four, or more than seven decimal places. The following is the rule by which the power of a table of loga- rithms is to be judged. The number of places of figures which may be obtained in a result de- rived from any table of logarithms, is the same as the number of decimals to which the logarithms are carried. But towards the end of the table, the last place thus obtained cannot always be depended upon within a unit. We shall proceed to the description of the arrangements of several tables, such as are most likely to fall in the reader’s way. EXAMPLES OF THE PROCESSES I. The tables which run to seven one form, of which the following is a places of decimals are all arranged in specimen. No. 0 ON ate ares ORY aa RG Ct Ol ae 38 Diff. 14550} 6580114)/0209\6305/0400/0496|059 1|0687|0782|0877|0973 95 1 1068)1164]1259}1355}1450|1545)1641/1736|1832)1927 2 2023/2 118/2213/2309/2404|/2500/2595/2690/2786|/2881 9172/926 7|9363|9458/9553 9648, 9744/9839/9934/0029 0125/0220/0315/0410/0506 0696/0791/0886 0982 1077)1172)1267/1362) 1458 9 8696|879 1/8886|8982|9077 4560 1) 6590601 OMWIMS OER WY— rm ice) The first column contains the first four places of the number, and over the head of the page is the fifth place of the number. The first three places of the logarithm (which throughout the specimen are either 658 or 659,) are not repeated with every logarithm, but only inserted at (or as near as may be to) the place where a change of the third figure takes place. But the best way to explain this table will be to destroy arrangement and abbreviation, and be- gin to write it down at full length. The student must account for every figure of the following out of the specimen. The characteristic need not be inserted, as what we here take out is merely the decimal part of the logarithm. Log. 45500 °6580114 Log. 45501 — *6580209 Log. 45502 6580305 Log. 45503 6580400 Log. 45509 °6580973 Log. 45510 °6581068 Log.45511 —*6581164 Log. 45602 *6589839 Log. 45603 6589934 Log. 45604 “6590029 4 “gangeor Log. 45605 *6590125 It would break the page to show that 658 becomes 659 in the middle of it; and various methods are used to remind the computer that the change has taken place. In different works, the line 4560, after the change, is varied thus :— 0029 | 9125 | &e. or 0029 | 0125 | &e. or 0029.| 0125 | &e. all serving toremind that the first three places must be looked for zmmediately below, instead of more or less above, the line of the last four. The column marked Diff. (for dzffer- ence) shows how to find the logarithm of a number of six or seven places of figures. For instance, what is the lo- garithm of 4551132? Take out the ‘decimal part of log. 45511; to this add what comes opposite to the sath place in the column Diff. ; (the sixth place is 3, and 29 is opposite to 3 in column Diff.) ; add the nearest number of tens in the number opposite to the seventh place (the seventh place is 2; opposite to 2 in col. Diff. is 19, nearest number of tens, 2 tens) and the result is the decimal part of the logarithm required: thus— Log. 45511... °6581164 3. 29 2 2 Log. 4551132 *6581195° Log. 455°2008 = 2°6582031 Log. 45603°97 = 4'6590027* Log. °4560444 = 1°6590071 In the earlier part of the tables, where columns of differences occur more thickly, several for the same line of logarithms, it is almost immaterial which is used; but for safety, take that column of differences which is headed by the difference between the logarithm taken out and the next following it. To find the number corresponding to a given logarithm, look in the table for the decimal places, which are nearest below those of the given logarithm ; take out this logarithm, and the jive places of the number, subtract the lo- garithm taken out from the given loga- rithm, Look in the second column of the * The change in the third figure takes place in the process, OF ARITHMETIC AND ALGEBRA. 39 _ differences for the number next below the result of subtraction just found, opposite to it will be found the sixth place of the number. Subtract the number used in the second column of the differences from the result of sub- traction above-mentioned, annex a ci- pher, and repeat the process with the column of differences, taking the near- est this time, whether above or below; the result is the seventh place of the number, For instance, what is the number to the logarithm 1°6582554 ? Subt. and annex 0 60 Opp. to 6 is 57, the nearest. The first five places are 45525, the sixth is 5, and the seventh is 6, so that 4552556 is the number required; and because 1 is the characteristic, there must be two places before the deci- mal point; that is, 45°52556 is the answer. The following useful numbers are mostly taken from the list at the end of Mr. Babbage’s logarithms. They will °6582554 serve as exercises, either in taking the Nearest log. below logarithm to a number, or the con- No. 45525 6582500 © verse. 54 Opposite to 5 is 48 No. Log. Circumference of circle (diam. being 1) 3°141593 *4971499 Area of circle (do. do. jie ° 7853982 1°8950899 Content of sphere (do. do.) . 5235988 1°7189986 INTL SCCONUS TLE AON tess ctin: ah ouy sot ae 1296000 6° 1126050 No. of ares of 1” inthe radius . 206264°8 5°3144251 Wo. of ares of 1’ in the:radius.’ .> .-. - 3437° 747 3°5362739 No. of ares of 1° in the radius 57°29578 1° 7581226 Base of Naperian logarithms ._, 2°718282 *4342945 Modulus of common logarithms *4342945 1'6377843 Metres in a toise Limited Co aiehs s 1°94904 °2898200 PY ALOSOie a SOBER aR. ol Santen 6 BPS Toot *3286916 Peet ited folser be. ye rts Paha crise cee Ly 6°394593 °8058129 Yards in a metre 1°093633 *0388716 PECL Uh a IGETOs Saul vhe le tee ah esl le 3° 280899 *5159929 PEGhes IME CClOMEr Se etek Demy ots eo) ve 39°37079 1°5951741 Feet in a Frenchffoot) Aci. (ele ois 1°065765 * 0276616 Acresiimanare.(French) 008.6 0 rey *02471143 2°3928979 Lbs. troyina gramme. . . . . . ~ ‘100268098 3°4282998 Lbs. avoir. in a gramme 00220606 3°3436173 Cwistin a. kilogramme! ). {ieee us °0196969 2°2943993 CEL whee aR Ge (7b El gs) Sl arlye Eig 0b RPM ieee *2200969 1°3426139 Seconds in 24 hours 86400 4°9365137 Diurnal acceleration of stars in mean en 235°9093 OS 727451 seconds 142-5 ; Common tropical enn in mean isola days 365° 2422 2°5625810 Grains in a cubic inch of water (barom. 30 Mola term. 62.hanr) «svn a . Tek } 252° 458 24021891 Inches in the pendulum, which vibrates seconds in a vacuum in the latitude tf 39° 1393 1°5926130 ECOL ete tse) Be ea eae ' II. The second set of tables, which it will be worth while to describe, has five places of figures in the logarithms, and four places in the number, with a difference to find a fifth. We have not deseribed logarithms of six places, partly because they are arranged much in the manner of those which have 40 seven places, and partly because tables of six places are of comparatively little use. For most practical purposes out of astronomy, and for very many of the details of calculation connected with the latter science, five places are amply sufficient : and where five are not suf- ficient, seven are much more frequently wanted than six; besides which, the arrangement of most tables of six places which we have seen is so de- fective, that those of seven are, in our opinion, more easily used. The best tables of five places (though with a very singular and awkward de- fect, presently to be noticed), are those of Lalande*. The following is a spe- cimen :— Lo. a 30” | Nomb. Logarit. ccs LW 1290 3°11059 a 1291 3°11093 o 1292 3°11126 BMS hI fa 1293 3°11160 33 1294 3°PD193 a 1295 3°11227 Dini olds Bee eas ee 1296 3°11261 34 1297 3°11294 1298 3°11327 he defect alluded to is the charac- teristic, which is inserted as if the lo- garithms of whole numbers of four places were always required, to the exclusion of all others. Thus, though the characteristic above given is cor-- rect for the logarithm of 1292, it is not so for those of 129°2, 12°92, &e. The best way for the student who uses this work, is never to think of the characteristic as anything but an addi- tion to the boundary line; that. is, to look upon the numbers as separated from the decimal part of their loga- | rithms by a fanciful boundary, like EXAMPLES OF THE PROCESSES To find the logarithm of any four places, simply look in the table and choose the right characteristic. Thus: Log. '1295 = 1°11227 To find the logarithm of *12956, take the difference which comes next under 11227, namely 34; multiply it by the new figure 6, but instead of writing down the first place, carry the nearest number of tens to the next place. Say, 6 times 4 is 24, carry 2; 6 times 3 is 18 and 2 is 20. So that Log. °1295° = 1°11227 6 20 Log. °12956 = 1°11247 Log. °1295° = 1°11227 7 24 Log. °12957 = 1°11251 To find the number to a given loga- rithm, take out of the table the decimal part next below the given decimal part, and the four places opposite to it. An- nex a cipher to the difference, and divide by the number in the column of differences, taking the nearest quotient of one figure. That one figure is the fifth figure of the number. For in- stance, what is the number to the loga- rithm 2°11178 ? 2°11178 "11160 33) 180 (5 012935 8°11153 11126 34) 270 (8 “Ans. 129280000 nearly. We cannot fill up the remaining places out of this table, and must place ciphers instead. The real number to the log. 8°11153 is 129279600°09 very nearly. The numbers given in page 39 may be made exercises; but the nearest five significant figures of the number must be taken, and the nearest five decimal figures of the logarithm will be found. Example. What is Log. 3°1416 1293 Ans. Given Log. 1292 a4 Log. 3°141° —-0°49707 3 instead of simply 6 8 3 Ba No S. Log. 3°1416 0°49715 Ans, * The title-page of the best edition is as follows:—‘ Tables des Logarithmes pour les nombres et pour les sinus. Avec les explications, &c. &c. &c. Edition Stéréotype gravee fondue et imprimée, par Firm1N Divot, A Panis, &c, 1805. (tirage de 1831).” The last four words should be particularly looked at, OF ARITHMETIC AND ALGEBRA. III. There are logarithms of four places on the table givenin the Trea- tise on Arithmetic and Algebra, which are sufficient for many _ purposes. These tables are arranged somewhat after the manner of those of seven Log. 16°8 -+ = 1°2253 4 11 1°2264 Log. 16°84 The number toa logarithm might be found by the reverse process. Thus: 1°2687 ieee 2672 6 or 7 15 Ans. ‘1856 or °1857 But these tables are accompanied by an anti-logarithmic table, in which the numbers and logarithms change places; so that a number is found from its loga- rithm by the same process as that which finds the logarithm from the number. For instance, in the preceding example, 268 + 1854 7 3 1°2687 is Log.of *1857 Given Log. The table of anti-logarithms is more trustworthy than the inverse process with the table of logarithms. Given No. 16°3 Log. 16 -- = 1°204 D 26 3 8 Log.16°3 = 1°212 Given Log. 2°308 30 +. 200 D4 8 3 2°308 0203 41 places, with the exception of the co- lumn of differences being placed hori- zontally, with a common heading. A few examples of the method of taking out logarithms will suffice : Log. 1°69 +» = 0°2279 1 3 Log. 1691 = 0°2282 IV. We subjoina table of logarithms and anti-logarithms to three places only ; partly because there is considera- ble power in such a table, and partly be- cause it will be a guide to the beginner in consulting larger tables, as he may thus (while new to the subject) find out to what part of the larger table to turn, in either of the operations of taking out logarithms or numbers. In the following table the differences between successive logarithms are placed in smaller figures in the inter- mediate space. As these tables are only an abbreviation of those already described, the first gives the log. of two places to three places ; the second gives the proportional parts of the differ- ences ; the third gives the number of a log. of two places to three places. When the difference is 12 or under, the proportional part can be found at once by the process described in speaking of tables of five places : Given No. 109 Log. 10 -- = 1°000 D 41 9 37 1°037 Given Log. 1°496 49 +. 309 D7 6 4 1°496 31°3 V. Finally, we recommend the student to commit to memory the following table of logarithms to two places: No. Log. No. 00 30 48 Log. No. Log. 60 7 85 70 90 18.ctp Beye 2H 42 EXAMPLES OF THE PROCESSES LOGARITHMS. 1 2g 3 4, 5 6 7 8 9 0 000] O 301) O 477) O 602) O 699} O 778 0 845) 0 903 0 954 7 6 5 5 4) 21 14 ll 9 1 041| 1 322) 1 491) 1 613) 1 708] 1 785] 1 851) 1 908] 1 959 ' 38 20 14 10 8 ve 6 i 6} . 5 2 079| 2 342) 2 505} 2 623] 2 716| 2 792! 2 857) 2 914] 2 964 vee: 20 14 10 8 * 6 5 4 3 114| 3 362! 3 519| 3 633! 3 724) 3 799] 3 863} 3 919] 3 968 32 18 12 10 8 vi 6 5 5 A 146! 4 3801 4 531] 4 643] 4 732] 4 so6| 4 869] 4 924] 4 973 30 18 13 10 3 7 6 5 7 5 176| 5 398] 5 544! 5 653! 5 740] 5 813} 5 875} 5 929) 5 978 28 17 12 10 8 7 6 5 4 6 2041 6 415| 6 556| 6 663! 6 748] 6 820) 6 881} 6 934) 6 982 26 16 12 9 8 6 5 6 5 7 9301 7 4311 7 5681 7 6721 7 756| 7 826) 7 886) 7 940| 7 987 25 16 12 9 7 4 6 4 4 8 255| 8 447| 8 5801 8 6811 8 763) 8 833] 8 892] 8 944] 8 991 24 15 ll 9 8 6 6 ia) 5 9 279| 9 462] 9 591) 9 690| 9 771) 9 839] 9 898! 9 949] 9 996 22 15 11 9 *4 6 5 5 4 PROPORTIONAL PARTS. 41 38 35 32 30 28 26 25 24 22 21 20 19 18 17 16 15 14 18 “i fre iT RE UE i er ee er Ce Miya > adBiGee FO 6 pOwibes bem 4) 40 45 24) 8 omega ee SD. Fe LPO Dia A SNR) 7 Bin 6 Fb eb et oe Bde AIG AGI 1B IZ LP 10. AOO, 9 8 Ble Been 8 Sy 2) 19 18 16 15. 14°13 13 12 11 1110 109. 9 SB read 6 | 25 23 21 19 18 17 16 15 14 13 13 12 11 11 10 10 9 8 8 7129 27 25 22 21 20 18 18 17 15 15 14 13 13 12 11 11 10 9 8133 30 28 26.24 22 21 20 19 18 17 16 15 14 14 13 12 11 10 937 34 32 29 27 25 23 23 22 20 19 18 17 16 15 14 14 1312 41 88 35 32:30 28 26 25 24 22 21 2019 18 17 16 15 14 13 ANTI-LOGARITHMS. a0) ‘] 2 3 4 5) ‘6 | oF ae i 0 100} O 126] O 158} O 200] O 251] O 316} O 398] O 501] O 631; O 794 1 102 1 199 ] 162 1 204 1 o57/ 1 324 1 407] 1 513] 1 646} 1 813 | 2 105| 2 139 2 166 2 200] 2 263] 2 331] 2 417| 2 525] 2 661] 2 832 8 107 3 135 3 170| 3 214 3 269 3) 339 3 427| 3 537| 3 676| 3 851 | 4 1101 4 13s] 4 1741 4 219] 4 2751 4 347] 4 437] 4 550] 4 699) 4 871 : ; 1 5 178 5 904 5 989 5 355 5 447| 5 562) 5 708] 5 891 1 6 115 6 145} 6 182 6 999 6 288 6 363 6 457) 6 575| 6 724) 6 912 17 117] 7 1481 7 186 1 934| 7295/7 372 | 468] 7 589| 7 741| 7 933 is 120| 8 151 8 191 8 240 8 302 8 380 8 479| 8 603] 8 759] 8 955 9 123 0 155 o 195 9 245 9 309 2 389 9 490| 9 617| 9 776| 9 977 3 5 mo OoOnNF Oo & ao no = OF ARITHMETIC AND ALGEBRA. 43 Szorion 7,—Application of Logarithms worked at Length. In most of the following examples, we shall use the tables of seven places ; those who employ smaller tables can produce the same result, as far as their tables go. The following is an instance of the way in which the same.question must be treated, according to different tables: _ What is *1234567 x 26813°92? 1. With tables of seven places : Log. °12345.. = 170914911 6 212 7 25 Log. '1234567 = 1°0915148 Log. 26813.. = 4'4283454 9 146 me 2 3 Log. 26813'92 = 4°4283603 10915148 Add 375198751 33103... 5198674 77 5 66 8 110 Ans. 3310°358. 2. With five places: or what is *12346 x 26814? ‘Tog. 21234. =" 1709132 6 21 Log. °12346 = 1°09153 Log. 2681. = 4742830 4 6 Log. 26814 = 4742836 1°09153 3°51989 3310° 51983 13) 60(5 ANS. 3310°5. 3. With four places : *1235 X 26810: Tog, “125. ~*1" 0899 5 ne Log. +1235 1°0916 Log. 26800 474281 1 2 Log. 26810 4°4283 1°0916 1919;% 3304 9 7 "5199 3311 Ans. 4, With three places: *123 X 26800: Toe. 212% 5114029 (Do fi 3 11 "123 Log. 1°090 Log. 26000 4°415 8 13 Log. 26800 4°428 , 1°090 3.518 Blas... 824 8 6 3°518 3300 Ans. In future we shall give the logarithms to seven places, but without going through the detail of using the table of differences to find the sixth and seventh places, either of a number to a loga- rithm or of a logarithm to a number, except in a few particular cases. . 1 Question 1. Find 1034°9 Log. 1 = 0°0000090 Log. 1084°9 = 3°0353897 000921744 -4°9646103 Question 2. Find:/°1 and °/97.65625. Log.*1 2) 170000000 ~ 1°5000000 31622 4999893 107 " 96 8 110 or V/°1 = °3162278 Log.97°65625 5) 1°9897000 2°5 Ans. 3979400 44 In the last result, the exact coinci- dence (to seven places) of the answer with 2°5 may induce a supposition that 97°6562Z5 is the exact fifth power of 2°5, which is really the case; but no- thing can be inferred from the tables, except that the fifth root of 97°65625 lies between 2°4999995 and 2°5000005. It might be 2°499999576. ccaseee or 2°500000214...... and the answer of the tables would still be 2°5, Question 3. %/(32°92416 x 10°27251)® 1084°9 1°5175147 1°0116766 2°5291913 6 5) 15°1751478 ee 3°0350296 Log. 1084°9 3°0353897 Ans. °9991712 1°9996399 Question 4, Find a fourth propor- tional to 1234, 2345, and 3456 ; or find 2345 X 3456 + 1234: Log. 32°92416 Log. 10°27251 Log. 2345 3°3701428 Log. 3456 3°5385737 6°9087165 Log. 1234 3°0913152 Ans, 6567°518 3°8174013 We shall hereafter give a more expe- ditious way of solving this question. Question 5. What is the thousandth power of 2? Log.2 = *3010300 1000 Multiply 301°0300000 Now, this is the logarithm, as nearly as our tables will tell, of 107151900000..... the number of ciphers being 294; that is, apparently, the thousandth power of 2is a number of 302 places of figures, the first seven of which are 1071519. But it must be recollected that a thou- sand times *3010300 is 301°0300, and that we only annex four more ciphers because we do not know with what figures to fill up the vacant places, We cannot, therefore, depend upon more than four places of the result, and should say that 2 is a number of 302 figures, of which the first four are 1071. If we would have the first seven EXAMPLES OF THE PROCESSES places’ correct, we must go to a table of ten places at least. This gives Log.2 = °3010,299957 Log. 2 = 301°0299957 1071508 0299922 35 so that the first seven figures are 1071508. Let us here observe, that by mere inspection of a logarithm, we answer questions which would take years of calculation. For instance, from the above logarithm of 2, we see that the tenth power of 2 has 4 figures (3-41) ; the hundredth power has 31 figures (30+1); the millionth power has 301,030 figures, and soon. Hence the simplest method i” theory, of caleu- lating a logarithm to seven places, is by the following formula :— { No. of fig. in ten- Log. x =- millionth power of x} we ten million but this, of course, would be practi- cally impossible to use. Question 6. What whole number is that which has 256 places of figures in its 70th power. The logarithm of that 70th power must be between 255°000.... and 255°999.... that is, the logarithm of the number itself must lie between 255°000.5 5. 255°099; 254 —— and ———_——_— 70 70 or 3°6428571 and 3°6571428 Answer: All whole numbers between 4394 and 4540, both inclusive. Question 7. What is the value of a/ 5138 "S01 Log. 138 3)2°1398791 *7132930 Log. 5 "6989700 Log. 5138 1°4122630 Log. ‘01 5)2°0000000 1°6000000 1° 4122630 2)1° 8122630 Ans. 8° 056224 9061315 Operations with logarithms may be divided into—1. Those in which a num- ber need never be found to a logarithm OF ARITHMETIC AND ALGEBRA, until the end of the process ; 2. Those in which numbers must be found to logarithms as a subordinate part of the process. All the instances hitherto given, and all which involve only mul- tiplication, division, raising of powers, and extraction of roots, fall under the first case; while all which contain ad- dition or subtraction fall under the second. For instance, to find J'V4 4. OWS : oi S32 faa we must first find V5, then 4, then make the addition indicated, and find the square root of the sum. Log. 4. Log. 5. 3)* 6020600 3)* 6989700 "2006867 2329900 1°587401 1°709976 1°709976 3°297377..6 2)°5151686 Ans. 1°809607 9575843 Question 8. / (01 C01 -01)) ? Log. ‘01 10)2*0000000 1°8000000 2° 0000000 10)3°8000000 1°7800000 2° 0000000 10)3" 7800000 Ans, °599791) 1°7780000 Repeat the process until the tenth root has been extracted seven times, and show that the result will then be very nearly equal to the ninth. root of *01. Question 9. Supposing the earth to be 7916, and the moon 2160 miles in diameter, how many times does the bulk of the former contain the latter ? [Spheres are to one another as the cubes of their diameters; that is, if one diameter contain another x times, the sphere on the first contains that on the second x 2 x times.] The question is, what is (7916 — 2160)? ? Log. 7916 3°8985058 Log. 2160 3°3344538 0°5640520 3 Ans, 49°22163 1°6921560 Answer—About 49+ times. 45 Question 10. What is the number of cubic miles in the earth and moon, the diameters being as in the last question ? [To find the cubic miles in a sphere, multiply the cube of the diameter by the cubical content of a sphere of one mile in diameter, page 39.] Log. 7916 Log. 2160 3°8985058 3° 3344538 3 3 11°6955174 10°0033614 1°7189986 1°7189986 11° 4145160 9°7223600 Answers nearly Question 11. To how much will 15/. 7s. 34d. amount in fifty years, at 3 per cent. compound interest; or what is eee at x, (1°03) The sum mentioned is £15°364. 259726400000 | } 5276671000 Log. 1°03 0128372 50 “6418600 Log. 15°364 =: 1° 1865043 67°354 1° 8283643 Answer—£67 7 1—very nearly. Question 12. How many feet are there in 867°41 metres [page 39, log. No. of feet in metre = *5159929.] Log. 867741 2'9382244 Log. (feet in metre) —°5159929 3°4542173 Answer—2845° 885 Question 13. Taking it for granted, as is proved in a higher branch of ma- thematics, that when z is a large num- ber, the product Pe M2 KS) Ku deus ee is very nearly equal to x (@-1) X & W6-2831851 XB Xoo ; 6°2831854 X @ X)5-7) 0818 what is (nearly) the product of the first thousand numbers ? is it greater or less than would be obtained by substituting the average for every one of the num- bers, and how many times does the greater contain the less? Also how many figures are in each product ? [The average of 1, 2, 3...... 1000 is 500°5, and the products to be com- pared are therefore 1X2X3%X%.e006-X% 1000 & (500°5) 0) 46 Log. 6° 283185 Log. 1000 *7981799 3° 0000000 2)3°79 7981799 1°8990899* 3° 0000000 *4342945 2°5657055 1000 2565° 7055 1°8991*. 2567 °6046 Hence the product of the first thousand numbers contains 2568 figures, of which the first four are 4023; and the best approximation we can make is— 4023000,...(2564 ciphers). Log. 500°5 2°6994041 1000 2699°4041 2567°6046 131°7995 It appears that the second product has 2700 figures, the first four of which are 2535: itis incomparably the greater of the two, and contains the first a number of times, having 132 figures, the first four of which are 6302. As some further examples of the preceding formula, let [a] signify the product of all the numbers up to & inclusive; Log. 1000 - Log. 2°718282 then— Log. [1010] = 2597°6284 Log. [1020] = 2627°6952 Log. [1030] = 2657°8046 Log. [1040] = 2687°9561 Log. [1050] = 2718°1493 Log. [1100] = 2869°7278 Log. [1150] = 3022°2933 Log. [1200] = 3175°8028 Question 14. What whole power of 2 is nearer than any other to 100,000,000? That is, how many times does the lo- garithm of 100,000,000 contain the lo- garithm of 2? Log. 2 Log. 100,000,000. "30103) 8° 00000... ..5,4 (26°57 Ans.—The 27th power. To get examples by which thestudent may ascertain whether he has acquired the highest degree of accuracy in taking out logarithms, &c., the veri- fication of cases such as those in page 27 (rule 2) will be useful. For in- stance :-— Question 15. Verify to seven places EXAMPLES OF THE PROCESSES. 5 of figures, (if the logarithms to seven, places will serve) the equation pe cae Z NEE dou 1h i=aun 2a tee 18—‘* 1} Log, 18. Log. 11. 2)1°2552725 2)1° 0413927 *6276363 °5206964 4°242641 3°316625 3°316625 7°559266 Sum | its log. *8784795 0°926016 Diff. | its log. 1°9666185 0° 8450980 which is correctly the logarithm of 7. At the beginning of the tables (1000...) an alteration of a unit in the seventh figure of the number makes an alteration of 4 units in the seventh figure of the logarithm; so that two logarithms, which differ only in the seventh decimal, .by less than 4, are for every practical purpose the same. But in the last half of the tables, a unit of difference in the seventh figure: of the number causes less than a unit of difference in the seventh place of the logarithm, which renders the tables not so safe in the latter part as in the former. To illus- trate this, we form the following table from the extreme end of the table to seven places, repeating only the figures which change. No. Dec. Part of Log. 9999900 90 os eceee eoevcee ecees Conoanrhodanre From this it appears that *9999960 may belong to 999990, followed either by 6, 7, or 8, so that the number can- not be found within two units in the seventh place. But this is the extreme point; and, generally speaking, the re- sults may be depended upon within one unit in the seventh place, which is always more than sufficient for prac- tical purposes. Question 16. What is the value of a in the equation? (20)* = 100 * It is useless to retain more than four places of this, OF ARITHMETIC AND ALGEBRA. 47 This is the same as asking, what isthe in ten years? That what is the solu- logarithm of 100to the base 20? Taking tion of the logarithms of both sides, we have I +2)" = 2 Log. 20” or a x Log. 20 = Log. 100 v= "9 — 1 = 1071773 or 7°177 per cent.; that is £7 3 64 _ Log. 100 2 per cent. Log. as 1°537244 90 ~ 1°30103 In working questions of compound interest. for long periods of time”, it is sometimes necessary to have certain Question 17. At what rate of com- logarithms to more than seven places. pound interest. will money double itself The following will be sufficient. No. 10025 10050 10075 10100 10125 10150 10175 10200 10225 10250 10275 10300 10325 10350 10375 10400 10425 10450 10475 10500 10525 10550 10575 10600 Rate per cent. in Dec. part of Log. which this Log. is used, 00108 43813 00216 60618 00324 50548 00432 13738 00539 50319 00646 60422 00753 44179 00860 01718 00966 33167 01072 38654 01178 18305 01283 72247 01389 00603 01494 03498 01598 81054 01703 33393 01807 60636 01911 62904 02015 40316 02118 92991 02222 21045 02325 24596 02428 03760 02530 58653 Rie WIR RIE ee ee eee cael wowwowrnnd bd vo HRYOo bole Fl Alto bo Alto wile Pl Rio ele Ble Cs) ad Darran k & & Question 18. What is the amount of | This may be done by the following one farthing, for 500 years, at 3 per table: cent. compound interest ? : ; 1 | 023 025 851 One farthing is £.°001041667, and 9 ate 051 702 the quantity to be found is 3 | 069 077 553 £. (1°03) X *001041667 ; oy Os See Log. 1°03 pines7248t, 1 Mate a nt eee etait 7 | 161180 957 ) 6° 41861235 8 | 184 206 807 Log. *001041667 3°0177286 9 | 207 232 658 2731°121 = 3° 4363410 This table is intended to abbre- Ans. £2731 2s. 54d. viate the operation of multiplying by Question 19. 2°3025851, and its use will be evident Given the common from the following examples. What is, logarithm to find the hyperbolic or first, the Naperian logarithm of 56? -Naperian logarithm. ; * Such, for example, as Dr, Price’s celebrated problem about a farthing put out to compound interest at the beginning of the world. 48 EXAMPLES OF THE PROCESSES | Common Log. 56. in the table with all its places, attend- In table, we findoppo- 1° 7 4 8 1 8 8 0 site to 1 la Baia Sea | RR SHES Ss BS NE S| 7 1 Be a Re eB 0 ees ae 4 0 9929 Woe ot 4 8 LD as ee ont 1 0.2. 43:90 .3 8 Ivigued og 8 bo BY 4 De GO a el ay. Make seven decimal places, and the answer is 4'0253517, the Naperian logarithm of 56. What is Nap. Log. 9828 ? Common Log. 9828. 3°9924651 069077553 29 7 2°39 646 20 Fee 0460 52 pots ap peeiee 115 0 2 91929907 Ans. 9°1929907. Examples for practice : Number. Nap. Log. 3°141593 1° 1447299 2349 7°7617450 156°3 5° 0517778 When there is no characteristic, use one place less, and make seven places. When there is a negative character- istic, neglect it, and proceed as in last sentence; but subtract at the end the number opposite to the characteristic ing to page 35. What is Nap. Log. +008? Common Log. °008. 3*9030900 20723266 069078 2072 20794416 069077553 —— 51716863 Ans. 5°1716863 [N. B. As a check upon this rule, remember that the Naperian logarithm must be something more than twice the common logarithm. ] Question 20. To reduce the Naperian logarithm to the common logarithm, use the following table in the same manner : 0434 2945 0868 5890 1302 8834 1737 1779 2171 4724 2605 7669 3040 0614 3474 3559 3908 6503 The Naperian logarithm being 9°1929907, what is the common loga- rithm ? CONankwrnore 9° 1929907 39086503 0434295 390865 08686 3909 391 3 39924652 Ans. 3°9924652 Section 8.—Examples of the Application of Logarithms for Practice. Before proceeding to give any exam- ples, we shall explain why we have deviated from the usual practice, and in a manner which some of our readers will consider rather singular. In work- ing rules by examples, which are pre- sumed to be quite correctly answered in the book, the student is apt to work by the answer—that is, to look at the answer from time to time, and judge, or at least guess, whether he is proceeding correctly. Very few have the resolu- tion to shut the book, and not look at the answer until they have produced theirown. The consequence is, that no confidence is gained, and the student has to learn how to be independent after he has left his elementary trea- tise, and has to solve questions which occur in practice. To give no answers at all, would be depriving him of an assistance which, to a certain extent, is useful, and even necessary. We have therefore made some figure or figures, or positions of the decimal point (per- haps many or all, but the student must find this out), intentionally incorrect ; so that while there will be enough to OF ARITHMETIC AND ALGEBRA. assist the student who is disposed to learn how to shift for himself, there will be enough to perplex the one who has no assurance of being correct, ex- cept what he derives from the printed 49 the answer are found to differ from those here given, let the process be thoroughly re-examined, until the stu- dent is satisfied that he has obtained the correct answer.* answer. When a figure or figures of 441°5059 1 bce = 4°921610 eee 13052°62 1 vous = 13169°25 — = ‘01927535 9914449 : 51°88 611527 = +0772889 = + 903444066 79122°35 3 291°2 7* 466382 1 eens, ee 8 WE OOSR0 = 8°071025 66° 52304 [agg at Or Of A049 ©9299 iA DEE AES Last = 1'191654 l — (°4663077)2 100 + (8°09784)? = 12°86759 WV (629° 3203)? + (777°144)? = 999°9998 s/+00001215 = °003485685 027. = * 1653168 +27, = +5196162 */4°355 = 1°633137 3/436 = 7°582787 3 -_— (13°22869)2 = 48°11445 10/6 = 1°018] + gl8 a Liem, sonb ones? 7654°3 x 794 7/8 = 1°3859 ery OE Us Re ei 13/3348 . 92 = . as * 5 /1728 = 1°904169 / a 1°146156 21 123 (;) = 11°87322 ( = 31°69104 8 637 “32 "0537 (2 = 1°44378 (-)" = *982693 53 7 (52072) X /(°000734)? { 42666\ 765 \10 tee) Vee ae GRP ERS GB 4 ey ge Sia NAVs ‘v6 ) = 1°215695 \ 8/(°26 /2) = °596544 Be 1 3425 fey wi n/a absa 7) = 98°94819 APLC! 6 253 —— = 2016°014 132 (7°356)° \/ (3°25) = 144°5972 nt) A De * Many of the examples are taken, mututis mutandis, from Meier Heirsch’s Sammlung von Beispielen, For- meln, &c:, Berlin, 1816; others from trigonometrical tables, &e. iE 6 16 (466871)7 x (3576)5 I (996003) & (°0071)2 8/(21 + &/19) = 1°470075 EXAMPLES OF THE PROCESSES 1780845 8/5°03 + %/°2) = 1°792929 5/(9°921 — 32/5'02) = 1°261866 TINS .ogsgnes VOT NO)” psi pses 2/2 A, 1+ V3 /3 — 1 heed Ahh Bid hee ee ee a 5) Pa? ° 629520 55 (V5 + 1) : M5 n/5) 4 J/3 — 1 A vs- + Btls + 4/5) = *9986295 8 a/ 16 ‘ 3 ) (= + 51/278) _ 1 -ogagag 5/17 hae Deh cn Oke W301 8-02 03) = *4640688 What is the diameter of the sphere which shall have the same content as a cube of 21°16 yards in length? Ans. 26°25 yards. Find the number of cubic feet in a cube of 15 ¢nmches long. Ans. 1° 953. What is the diameter of a circle whose circumference is 25000 miles? Ans. 7958 miles nearly. What is the circumference of a circle whose diameter is 7958 miles? Ans. 250004 miles. _ What is the area of a circle whose circumference is 22 feet? Ans. 38°517 square feet. What is the area of a circle whose diameter is 15°25 inches? Ans. 20°2949 square inches. _ The surface of a sphere being four times the area of its largest circle, what is the surface of a sphere of 4°5 ie in diameter? Ans. 63°6174 square eet. SEecTION 9.—Arithmetical Complement. A sphere of six feet in diameter is painted at the rate of a halfpenny per square inch, what is the cost? Ans. £33188. 70 The diameter of a sphere being 7 feet, — what is the side of the cube of equal solidity? Ans. 5°64228. If a sphere be 3 feet in diameter, how long is the side of a square of the same surface? Ans. 5°31736 feet. The squares of the times of revolu- tion of different planets being as the cubes of their mean distances from the sun, and the mean distances of Saturn and Jupiter being in the proportion of 9538786 to 5202776, and the time of revolution of Jupiter 4332°585 days, what is that of Saturn? Ans. 10759° 22. The diameter of a sovereign being *87 of an inch, how many miles would 600,000,000 sovereigns extend, if placed side by side? As, 8238°64, Trigonometrical Tables: their Use in common Calculations. The arithmetical complement of a number is the number by which it falls short of the unit of the next higher denomination. It is abbreviated into Ar.co. Thus: PACH N) ie etl Dod a bes at Ar. co. 893 = 1000 — 893 = 107 Ar. co. °669 = 1 — °669 = °33I1 The lowest denomination considered isthe unit. Thus: Ar. co. °0094 = 1 — °0094 not ‘01 — °0094 The most expeditious way of finding the arithmetical complement is as fol- lows :—Begin from the left, subtract every figure from 9, up to the lowest stenificant figure, which subtract from 10. Repeat the ciphers at the end, if any, No. 156°142 *0013754 Ar. co. 843°858 *9986246 No. 1798000 4009000 Ar. co. 8202000 5991000 When there is a negative character- istic, add it to 9, instead of subtracting it from 9. No. 1°439 Ar. co. 10°561 3°108 12°892 2°33 11°67 OF ARITHMETIC AND ALGEBRA. The student should now practise taking out from the tables, not the loga- rithms there written, but their arith- metical complements, without first taking out the logarithms themselves. The operation above described can be correctly performed in the head, with a little practice. For instance, looking in the table, and seeing °6123180, he should say—6 and 3 make 9, put down 3; land 8 make 9, put down 8, &ce., up to 8 and 2 make zen, which will give *3876820. To subtract a number, add tts arith- metical complement ; the result will be too great by a unit of the kind which was usedin making the arithmetical complement, Thus, 9-4 may be thus found : 9 + Ar. co, 4 — 10 and subtractions may be reduced to the subtractions of single units from the results of an addition. In the following examples, the first is the common me- thod, the second the one just described. From 966813 9°66813 Take 3°44210 6°55790 6.22603 16°22603—10 From 2°30746 2°30746 Take 3°42815 12°57185 4°87931 14°87931—10 From 1°21769 = 1*21769 Take 2°30999 11°69001 0°90770 10°90770—10 The better way will be always to write, after an arithmetical comple- ment, the unit which must be sub- tracted, after addition has been substi- tuted for subtraction by means of that complement. - What is 1835 + 968 — 1036, and 21648 - 9763-144? 1835 21648 968 237 — 19,000 _8964— 10,000 856 — 1000 - 11767 22741 10000 11000 1767 11741 Required a fourth proportional to 117°1097, 17°36482, 9510°565: Log. 9510°565 3°9782063 Log. 17'36482 1°2396702 Ar.co. Log. 117°1097 7°9314071—10 13° 1492836 —10 Ans, 1410°209. The student may now try any of the 51 preceding examples, with addition of arithmetical complements instead of subtraction. But we recommend him rather to avoid this method, which is very subject to error, except in the hands of a practised computer. The trigonometrical tables have al- ready been described (for trigonome- trical purposes) in the treatise on that science (page 51). The logarithms there given are generally made too great by ten; that is, instead of the subtractive characteristic 1, we have the characteristic 9, &c. ; or, instead of subtracting 1, we add 9, which makes the result too great by 10. In trigonome- trical operations this is convenient; but principally because the extraction of roots very seldom occurs. Ifwe had, for example, to extract the square root of the sine of 46°, which we find in the tables to be *7193398, and the tabular logarithm of which is 9°8569341 (but, in reality, 1 °8569341), the following process will be wrong in the charac- teristic: 2) 9°8569341 A* 9284671 for the dividend being 10 too much, the quotient will be 5 too much; or, rather, the addition of the dividend being in- tended to be followed by a subtraction of 10, the addition of the quotient must be followed by a subtraction of 5. In extracting the cube root, the following process gives characteristic and decimal part beth wrong: 3) 9°8569341 3°2836447 for, the dividend being too great by 10, the quotient istoo great by 33, or 3°3333333, and must be set right by subtracting this. But, to reduce the result to the tabular logarithm, the logarithm should be made 20, 30, 40, &c., too great before dividing by 2, 3, 4, &c., as in those cases the results will severally be 10 too great. But, per- haps, the better way is to restore the proper negative characteristic, and pro- ceed in the way already described (page 37). What we have here todo with the trigonometrical tables is to observe that they may be considered as registers of the value of certain expressions, which, being already calculated, may be re- ferred to, and thus the trouble of fresh calculation saved. We shall proceed to explain the following day 9° ad EXAMPLES OF THE PROCESSES Sine, Cosine. Tangent. a Vi) a2 — a l-a V1 lee pol Bibs Nias /} — (2 a Mabe a a 1 mbes 9s ay pete a Vita la 1 a 1 Vita Vite e a2 —1 1 —— laiepeoass wae, az—1 a a 1 @—1 ae a a Vae—I By this we mean, that if. be the sine of an angle, or if we can find a in the table of sines, we find Vi—a? in the corresponding line of the table of co- sines, abel “Rea in that of the table of tangents: if a be found in the table of tangents, we have opposite to it cs nee a in, that of sines, and ——=-—- Va V1 +02 in that of cosines. If the tables only give logarithms* of sines, &¢., we must look for the logarithm of a, and we find the logarithms of the above quan- tities. To use this method with great accu- racy would give very nearly if not quite as much trouble as the common logarithmic process, but by mere in- spection a few places of any result may be obtained; so that when. very con- siderable accuracy is not required, cal- culation is altogether saved. For in- stance— a a being °6346, what is /]+ ge if the first be a tangent, the second is the corresponding sine: we look in the table of tangents (Hutton’s), and find as follows, under 32° 24’, that the tangent being °*6346193, the sine is * 5358268, so that °*5358 will be very near the answer. To get a nearer answer, we must use the method which we proceed to describe. Cotangent. Secant. Cosecant, V1 a 1 PaaS isan We ii a 1 1 Veoh att ie Coulee NTee Lire ) vie Via a 1 ' a AE Nye Deora a9 ae yy a Beh Naveen, Definition. When a number is to be found in a table opposite to another number, the second, by means of which we know where to go in the table, is called the argument. Thus, in finding the logarithm of 56, we enter the column of numbers with the argument 56. Inthe last example, we enter the table of tangents with the argument °6346. The result obtained we shall call the resultant +. Let a be the given argument not exactly to be found in the tables. Let M and m (M being the greater) be the arguments in the tables, between which a is found to lie. Let R and 7 be the corresponding resultants to M and m. Thus, Column of arguments.|Column of resultants. a|? mm Tr or the following— 7 a ane, M|R according as the arguments decrease or increase the resultant of the argu- ment @ 1s (a—m) (R—r) "ts M—m when arguments and resultants both increase or both decrease together; and (a—m) (r—R) ys Ee Se tS M-m * Hutton’s tables, which give sixes, &c., as well as their logarithms, are decidedly the best of which we know for the engineer or mechanic. There are more useful tables for the astronomer, to which we need not here allude. For logarithms of numbers only, we belieye those of Mr. Babbage to be best for general use, of all those which contain seven places, + There is no correlative to argument in common use. The reader may take his choice of resuléant, inference, extract, function, &c,; either of which would be better than no word at all. OF ARITHMETIC AND ALGEBRA. when one of the two increases and the other decreases. This process is called interpolation. In the above example, we are to find the sine to the tangent *6346000 (the argument). Looking in the tables, we find as follows— Arguments. Resultants. Tangents. Sines. m = 6342113 | 5355812 =r a = 6346000 | ? Wh =6346193 } 5358268. = KR Arguments and resultants increase to- gether. M —-m=4080 a—m=3887 R-r=2456 pees ee 2340 (nearest whole No.) 4080 . 9395812 2340 °§358152 Answer. This is within a unit, in the last place, of the truth. An unlimited number of examples of the preceding process may be ob- tained by taking an argument and re- sultant from the tables themselves, and finding the latter by means of those which come before and after. Sup- pose, for instance, that the cotangent of 19° 31’ had been erased or blotted so as not to be visible, and that it were required to fill it up by using the table of tangents, the process would be as follows :— Arguments. Resultants, Tangents. Cotangents, m = 3541186 28239129 = 7 a = 3544460 ? M = 3547734 28187003 = R Arguments increase and resultants de- crease. M-m=6548a—m=3274r —-R=52126 3274 52126 ede iat Tab PN TIN 6548 2°8239129 26063 2°8213066 Ans. which is incorrect by a unit in the sixth decimal place. This process may be applied to tables of logarithms ; or, in fact, to any tables. It is substantially what is done in find- ing the logarithm of a number inter- mediate 1o two numbers in the tables, or vice versa, and also in finding the logarithmic sine, &c. of an angle inter- mediate to two angles in the table, as the following examples will prove, if the student compare them with the usual process. 53 Required log. 6416958. We shall only consider the decimal part. Arguments. Resultants. Numbers. Logarithms. m = 6416900 | 8073253 = r a@ = 6416958 | ? M = 6417000 | 8073320 = R Arguments and resultants increase to- gether. M-m=100 a-m=58 R—r=67 98 X 67 100 8073253 ++ 39 = 8073292 Answer * 8073292 Required the logarithm of the sine of | 36° 18! 47/6, = 39 (nearest whole No.) Arguments. Resultants. Angles. Log. sines. m = 36° 18/ 0// O° 7 iso lausu i a& = 36°18! 47-6 | ? Mo = 56> 19.0" 9° 74729032 R Arguments and resultants increase to- oether. M—m=60" a—-m=47'"''6 R—r=1719 47°6 *% 1719 CaS (arene 1364 (nearest wh. No.) 9°7723314 1364 9°7724678 Answer. What is the angle whose logarith- mic sine is 9°9475008? Arguments, Resultants. Log, sines. Angles. im = 9°9474674 | 62° 23! 0! = # a = 9°9475008 | ? M.= 9:9475335.|62° 24' 0” = R Arguments and resultants increase to- gether. M-m=661 a—m =334 R—r=60 0 ooo = 30/"3 (nearest tenth) Answer 62° 23! 30! °3 What is the logarithm of the cosine OF B75! 9" * Sit Arguments, | Resultants. Angles, Log cosines. M579 6! Ol 9°7351345 = 7 O57 6 OR | 2 M = 57° 6’ 0” 9°7349393 = R Arguments increase and resultants de- crease. M—m=60"-a—m=9"'8 r—R=1952 . Ie: a = 319 (nearest wh. No.) 9°7351345 319 9°7351026 Answer. 54 What is the angle whose logarithmic cosine is 88852331 ? Arguments. Resultants. Log. cosines. Angles. m = 8°B849031 | 85° 36 0” = & a@ = 8°8852331 | ? Mve. 8"8866418 | 85°.35" 0" i Arguments increase and resultants de- crease. M—m=16387 a—m 3300 r—R=60" 3300 X 60 ~~ as = 12/9] (nearest tenth) 16387 Log. cos. 85° 34’ — log. cos. 85° 35! Log. cos. 85° 35’ — log. Log. cos. 85° 36’ — log. cos. 85° 37 An error of some tenths of a second in the answer is the consequence. In the tables of sines and tangents of angles under 2°, or of cosines of angles above 88°, the disparity of the differ- ences is so great, that it is useless to apply the above process; and it is usual, therefore, to give a separate table of sines and tangents for the first two degrees, in which the angles in- crease by seconds. In all other applications of the trigo- nometrical tables to calculation of com- mon algebraical formule, no very great advantage is gained, where much ac- curacy is required, by any tables which give the logarithms to minutes only. The reason is the length of the pro- cesses of interpolation. Where ex- treme accuracy is not required, the common tables, which go to minutes, are often advantageous, as in the fol- lowing instance :— To find /g2 4 ge let tan ¢ = ? then : a cosé, Examp.ie. What is /(92° 736)? + (64°018)?? Log. 64°018 = 1°8062343 Log. 92°736 = 1°9672484 Log. tan. 34° 37’ 9° 83898 Ve + = 9* 8389859 * 119672484 Log. cos. 34°37? 1° 9153846 + Ans. 112°68 2°0518638 The more exact process, beginning EXAMPLES OF THE PROCESSES 85° 36! — 12/1 = 85° 35! 47"°9 The accuracy of the preceding method depends upon the tabular differences continuing the same, or nearly the same, for the resultants of several suc- cessive arguments. This is the case for the most part throughout the tables ; but where it is not so, a small error is committed. For instance, in the last example, if we look at the table, we find— = ‘0016325 cos. 85° 36’ = *0016387 37! = *0016450 from the third line of the above, is the following :— Log. tan. 34° 36’ 51/""8 9°8389859 1°9672484 Log. cos. 34° 36’ 51"8 = 1°9153965 Ans. 112°6813 2°0518519 The following method of verifying any term in a table may be useful when an error is suspected, and no other table is at hand for comparison. Let 2 be the suspected term, and let A, B, C, &e., be those which precede, and a, 6, c, &c. those which follow; so that the table proceeds thus : “DE COBY Aion ae a hens When the tabular differences appear uniform in the neighbourhood of the term z, use the first of the formulee at the head of next page ; but where this is not the case, count the number of places of the differences which alter at each step, and use the formula opposite to that number in the next page. For instance, suppose we wish to verify the logarithm of the sine of 1° 12’, we have— A = 8°3149536 B = 8°3087941 @ = 8'3270163 b = 8°3329243 16°6419699 16°6417184 15 6 249+ 6295485 99°8503104 _16°6412989 266°2708474 C = 8°3025460 99° 8503104 c = 8°3387529 20) 166*4205370 16°6412989 8°32102685 8°3210269 in the tables. pa * 9 is written for 1, that this result may be too great by 10, as are all the logarithms in the table of tangents. + It is more convenient here to restore the real characteristic. OF ARITHMETIC AND ALGEBRA. 55 2 jz=3 (A+ a9) 12|)2=4{4 (A+ a) - (B + dt 34/2=4415A+a)-6 (B4+0)4+C+4+c} 5,6 | 2 = 7{56(A +a) - 282(B+4+8(C +9) -O4aA$ The trigonometrical tables may be 348 = 8¢ +A =,0 made to furnish a solution of equa- , tions of the second degree”, in cases Log.a@ 0° 4771213 where the coefficients are too com- Log.c 0°6020600 plicated to admit of easy multipli- Log. ac 2)1°0791813 cle the division. The rules are as ior pe eters: Gopi Log.2 *3010300 ‘anes Ar. Co. log. b 9°0969100 — 10 2 = . aac ; oe : ae is ae i : ch {Let sin é = —> Log. sin 60° 0' 0"°0 =9°9375307 — 10 3 aa + bx —c=0 aac 4 ax? — bx = ¢ = of Lettand = —— é=— 60° 0’ 0” °0 Then the two numerical values of # ye 8) 307' 0! OF" will be Log. tan 4 4 = 9°7614394 — 10 vom 7 Log. “ac = °5395907 a ‘ee Aye paceman cot < Ar. co. lug. a = 9°5228787 — 10 “6666667 19°8239088 — 20+ and the signs will be as follows :— 1. Both negative. 2. Both positive. 3. Greater numerical root negative. 4, Lesser numerical root negative. The instance on the opposite side will admit of easy verification in the com- mon way. The process which we have put in brackets is one of those little artifices of calculation, which occur in hundreds, but which it is impos- sible to reduce to rule, and which nothing but practice can teach. We have got the first root by means of the [2Log.tan$¢ 19°5228788 — 20 2 03010300 i ac ea: formula —— tan $4; and which is a °6666667, as far as seven places of decimals. [In fact, the real root is 2.] It remains to find the second root from the formula — cot 4 ¢,so that, in like manner as we found the first root from the calculation of Log. tan.44 + log. Vac + Ar. co. log. a — 10 we should proceed to find the second from . Log. cot. $4 + log. Nac + Ar. co. log. a — 10 But since the tangent and cotangent we have of the same angle are reciprocals, or cot. $4 = — > tan. 4 4 log. cot.44= — log. tan. }¢4 consequently the second formula is the same as log. Vac + Ar. co. log.a — 10 — log. tan 34 which, calling P the result of the first, is P — 2 log. tan. 34 and it is thus that we have proceeded. The preceding example, given merely for the sake of illustrating the rule, is one which might have been more easily done by the usual method. But the following is a case in which much trouble will be saved by the trigono- metrical method. a b Cc 1°0082 a2 + 6°4347 @ — 2°4566 = 0 * In this process we suppose the student to understand algebra, as far as the solution of equations of the second degree, + 19—20 gives the characteristic 1. 56 EXAMPLES OF THE PROCESSES Log. a 0° 0035467 Log. ¢ 0°3903344 Log. ac 2) 0°3938811 Log. “ae 0°1969406 Log. 2 0°3010300 Ar. co. log. & 9°1914717—10 Log. tan. 26°3' 561" 9°6894423—10 é. = 92628! 56" +4 i tee 3° 1 6B ea Log. tan. 3 4 9°3644973 —10 Log. “ac 01969406 Ar. co. log. a 9°9964533 —10 °3613193 ' 19°5578912—20 -18°7289946 —20 6°743675 08288966 The numerical roots being therefore °3613193 and 6° 743675, we find, by the rule of signs before given, that the real roots are *3613193 and —6°743675. To form examples of this method, let the student proceed as follows: choose - any two numbers for roots ; for instance, 1°308 and — ‘486, and any value of a, for instance, 2°709. Take the alge- braical sum and product of the roots ; that is 1°308 — ‘*406 °902 1°308 X —°486 = — °635688 Multiply these by 2°709, the chosen value of a, giving 2°443518 and —1°722079. Change the sign of the first ; then the roots of 2°709 x? — 2°443518 ~—1°722079 | should be 1°308 and —*486. The only way to obtain any security in the use of logarithms is to work many examples, beginning with simple cases. Repeated failures will take place at first; but the student will finally ac- quire that sort of habit which suggests the right method independently of rules. We shall now proceed to examples of algebraical operations, Section 10. Division of Algebraical Operations. Algebraical Reduction, Ad- dition, and Subtraction of Integral Quantities. The operations of algebra may be di- vided into two classes:—1. Those which present no forms distinct from the re- sults of ordinary arithmetic. 2. Those which require the use of the negative or other symbol in the manner peculiar to algebra. The operations of algebra may be subdivided again as follows :—1. Those which are usually left to the student in the higher parts of the subject. 2. Those which are usually inserted at length. Among the infinite number of classes of examples which might be chosen, we shall confine ourselves to those which it is most necessary the student should be able to perform, in order to read any work on the higher parts of algebra, or on the differential calculus. We suppose the student to gain the demonstrations of the rules here exemplified from some other source. I. Questvons tllustrative of the meaning of the fundamental symbols + and —, to be answered without rules. 1. By how much does a + b exceed a? By how much does a + 0 exceed a-—-1? By how much does a + 6 exceed a — mya — b,anda—26? 2. On what does it depend whether a+ bora-+t cis the greater? When a-+ 6 is greater than a +c, by how much does the first exceed the second ? By how much does a+ 2 6 exceed a-+ 6? 3. On what does it depend whether a+ 6 — cis greater than, equal to, or lessthana? By how much does a + 6 — c fall short ofa + 6+], and ofa+ b+c? F 4, On what does it depend whether a — bis greater than, equal to, or less than a — c? 5. Ifa lie between b and c, c being the greater, what must a be in order that c — aand 6 + a may be equal? 6. Ifa be greater than 0, and z less than y, which of the following must be true, which must be false, and which may be either true or false ? a-+ & is greater than 6 + y aty. pO ne ha ged ve Opi s nil tee glaee a+z islessthan 6+y aty . 5 - 642 OF ARITHMETIC AND ALGEBRA, 57 1l.— On the Use of Brackets. Distinguish between the meanings of— a-+ (6+ c) and a+6-+ ce a—{b —(c — dad) } a—-—- (6+ c) and @a- +e ae thy —e — d, a—(6-—-c) and a—b-—-c a— (b ~ c)— d and (gi) — 6 and a@ — (0 =e), a-—-(b6—c — d). 2(a + 3B) and 2a-+ 06. Il1.—Simple Reductions. s. How is an expression altered if 8a, and after- terwards 5 a be added to it ? Answer, 13 ais added to it, which is thus ex~ pressed— Establish the following, and express OR eta Oni Gy dere ee ye the question asked, in words, as above. BP -Bia bw 2 a. Ps Va i6g &e. > 2D 9 P+4a-—-a 4a+(P —a) Ge eintery P + $a@.+2a 45 a-- (P28 P—-ataz=P P+6a-—3a 6a+ (P — 38a) Pete ae = Pee &e. Sc. P—3sat4a=P-+a 4a —(a-P) 5a- (2a - P) &e. UG ie Eas tera In the last two sets of the pr rt st two pre- : 2 at. ae soe P a 20 4 ¢ ceding, point out the cases in which they are possible, or impossible, and why. Repeat the whole process with P—a, P+6a, &e. Give all the ways of expressing P + 3 a derived from the preceding. 1V.—General expressions derived from the preceding. P+tmatna=P+(m+nya P —- ma-+ na Ptma-na=P+(m—-n)a or or = P-(~™—™m)a =P—(m—-nv)a =P+(nm—-m)a V.—Simple Reductions including Fractions. 1 1 1 13 Fadia sanla 1 1 i 13 -a+-a+-a@=—aor g lees Toe a a a 13a pie war 1 OF arty Ge TR cI aah cag. BR ees Oe 2 y: eh ay, mnpekoks 7 7 ies LE MEY) DRA SCR — he Lome] haga lal Pek ky / Ne ne ies Ge VI.— Reductions in cases where the arrangement of the terms makes the opera- tions appear impossible, a — 2aisimpossible, but a— 2a+ 3 a is to be considered as a misplace- ment of a + 3 a — 24, which is2 a. 6a—l0da+4a=0 Q2a—-9a—3at l6G4=6a cy —Anyt+lry=72xy ppoptipatP ac — 3a’c + 100 a’c = 98 ac 11 m+4m—llymt l2m=—>m ol 58 EXAMPLES OF THE PROCESSES VIl.— Generalization of the two preceding Articles. (m + n)a (m— n)a (m+tn+ pa matna -ma—-na mMmatnatpa= i MLENXL—pxr=(M+n— P)xX tb re = (G4e- =e Wey oth ae n q S VITI.—Additions tn which subsequent Reduction is impossible. Add together a ++ b, Ca d, Bice ff, ta andab—cf. Answerra +b+ec— dte—ft+ab—cf. Add togethera —b+¢a¢e-1+ cf+ 6d,andp +q%. Answer,a — b6+e+ac-1+f+eft6d+ pt q’. IX.—Additions in which subsequent Reduction is possible. Add a — 6,a — c,¢ — 7, andc-+-6. Answer,a —~b+a-ce+eo—-7+4+ e+6,or2a—6b+c-—1. Adda + banda — b. Answer,a + b+a—bor2a. Add abe—-a+trvyt12—-p7p* -q 2abec—4p?—xy-_gt2+a 10p?> — 100 + 40a—-14ay 4q —4p?+abe X.—Ruie of Addition, Let the first term of each expres- sion be considered as having the sign + Annex all the expressions with their signs. Make such reductions as are prac- ticable. X1.— Subtractions in which From ataked — c. Ans.a—6b-+e. From 2 a@—c take 46 —d. Ans. 2a+d—c —4b, Add 13 2 + 20 vw? — 45 @ + ge — 2? + 502+ 32 30a? — 52+ 154 —8 22 — 07 + 2 — 10 le Further examples are deferred for the present. Rule of Subtration. Let the first terms of both expres- sions have the sign + Change the signs in the expression which rs to be subtracted. Annex the expressions, and make reductions. Reduction is impracticable. From p — q +r —stakez — pq. Ans.p—q+tr—-s—z+pgq. XII. Subtractions in which Reduction is practicable. From a takea —c. Ans.a -a+ Cc, Or C. From a + dbtakea — 6. Ans. a+ b6—a+,or2 06. From p — a take q — a. a—-qt+a,orp—4q. From ab+3a—-464+ 16 -2 Take 6ab— 12a—5b+122- 8 Ans. 15a —5ab+b424 — 132 From 6 a?6 - 12a+4-32—22a6 Take a26+ 100 —40ab+a-— 32 Ans.5ab— 134-993 + 18 a6 From 3 23? — 1827+ 62 — 150 Take 122% - 40277? +5 vw + 40 Ans. 22x27 —~ 92° + x4 — 190 Ans. p - From a—b+c—d+e-—ftyge—-h Take a—26+3 ct4d-i5et+f+g-8h (Ga+ 6) - (@+46) =46b6-ha @@a—- 6) —- (a—b)+c=e-}ha (2a+6) - (@-36) =a+45b Let there be any series of quantities, a, b, c, d, e, f, &e., and let another se- ries be obtained by taking the first from the second, the second from the third, the third from the fourth, and so on. Let this process be repeated with the second series, giving a-third, with the third series, giving a fourth, and so on, What will be the several series 2 OF ARITHMETIC AND ALGEBRA 59 Ist Series. 2nd Series, 3rd Series. a b—a e-2b+a4a b c —b d—2e+) c d—e e-2d+ec d e —d J-tert+d e Jf -eé g-2f+e Oi ee. hype! Biel f &e. &e. &e. 4th Series. 5th Series. d—3c+3b—-a e-4d+6c—4b0+a e—3d+3c—b f—-4et+t6d—4c+6 f —3e+3d—c g—4ft6e-4d+e ge —3f+3e-—d h—4g+6f—4det+d &ec &e. lst term of the Series. 6 f-5e+10d—-—10c+5b—a 7 g-—6f+15e—20d+ 15c—6b+a4 &c. &e. One way to form results which shall give examples of addition and subtrac- tion combined, is as follows :—Take any number of quantities at pleasure, sub- tract each one from that which follows, add all the differences and the jirst of the quantities together; the result should be the last of the quantities. For instance, let the quantities be a-b2a+6b—c,3a+26—Ae, and7a+ 66. From the second take the first, which givesa + 26 —c; from the third take the second which gives a+6—3c; from the fourth take the third, which gives da +4 6 -+44e. Add together the following— b—c¢ — 3c 6+ 4c which gives 7 a + 6 6, the last of the quantities first mentioned. As questions of mere addition and subtraction are of little use by them- selves, we shall now proceed to another point. Sgcrion 11, Exercises in the use of the Algebraical symbols + and — as distin- guished from Arithmetical* symbols. In whatever way the symbols’+ a and — a may be explained, the method of using them is as follows :— +(tayis +a +(-a) is —a —-(—-ais+a —-(+ ais -a or, tke signs produce +-, unlike signs —. at(-d =a-b= -b+ (+4 Th a Oat Dae Oh Oa b- (- a) ${p{ropp= te Po fa} ane @- b= 6 a) = —'b- (a) Th = Sorel Q—-3 => — 1 a-2a@=> - a4 Gs Rh ae G9 4—-7—8s& =- 11 a (Ye a ee i Sor et (2a + 36) — (4a + 76) = — (2a + 40) LG eS et ad Se ee Oe = BC) ile Aa rd — Wa — 3a-7b6—-4a+56=-—-a—20 ee ae Seo Ge eb +ax —-6b6= —a6b — HH = Pa +0 6 SS) a. ee SD 4 ahead tex Pm bale abed ci he Me ee Oe a Se a Oe d * The student may omit this section until he has read an explanation of the negative sign, 60 EXAMPLES OF THE PROCESSES —~3 x —-4= 4+ 12 1 RO -g ene ye GO Ki Os ae Ke =) fe Bh co KR Og a ax —-— 1 —« 2 a ie Pied A Cowle we) q-p x—y —(@—y yY—2 + a a ree Mash a Tho |. We sacra ee — a a + 0b a or AAR Nei RM, Ab NEF bere Ga tae ab ity 2) q—P Ly cy cy FIP AG we: lo ae eh fen eae L-y —~(y¥ -— 2) y—- 2x a-b+ec=a-(b-c) =atCe - J) = —- (6-a—c) =—(b-a +e= - (6 — 0) 4.4 = - (-a+t6-c¢) =at+(—-d)+e What suppositions will make the following expressions identically the same ? ave+bry+cy®+dyte ax—by+cz 3a%°9-4ry—2y¥+ 7y - 6 3U+ y+ 2 UnS.s ea b=—-4 Ce 2 Q=3 b= —1le=1 d=7 e=—6; | and the following— PO ge +r $27 -—-a+ 1 Hnswp 2 Sig it 0g Hen ig a ene TA y* t= — 27 x? — y” Sy 5 78 a y? ge= = y ao Ves y® Ub) = —) fe Which of the following pairs are the same, and which differ in sign ? — x§ and (— 2x)8 — x and (— a) x§ and (— 2)6 a7 and (— 2)i xa xX (— yt x (— 2)8 and (— a)? x (— y)t x 23 Change the sign of 2 in the following expression, that is, ; write — x instead of + xv, and + 2 instead of — 2. a+batecat+da® — — it becomes a + b(— x) +e(- 224+ d(- xp — — or, a-betea*—dat+—- Of the following expressions, the se- let the student find out which letter cond in each set arising from changing it is, and account for every change, the sign only of some letter in the first, a5 Ligh B © | tiie eva (inne I a a-be + t? _ ga ~ 2) 2. ax (ve — 2) + b2a%(¢ — 2°) nae oe —=\ @) bt OF ARITHMETIC AND ALGEBRA. 61 ax(@ — a) + b2* (2? + a’) —- 3. (Qaxt+ bye +4ac— + (Fate Dis km = Ob eh . eae ee Ea A. Go 0 Peta Oe Ame See or aN gare ba eee de ek xv When the signs of two or more let- ters are changed, the general rule is, every term changes sign in which an odd number of factors changes sign. Thus, in @+abc—akb—-BC if b and c both change sign, that is, if — 6 be written for 6, and — ¢ for c, &e., the term which changes sign is — a+b a-@ "a—b b—ab G— 2 ee a+) b+aa a—b+e 2. atb—-e oN Gin se OT ae i all jo + ay} rie C+ a Lt+ax RY + qa? 1 1 x? ee acb, or,acec, (in which an odd number of factors, c, c, b, change their signs) and the expression becomes @ +abe+acb — bc In the following examples one or more letters have their signs changed in the seeond and succeeding ones of each. The student must ascertain which letters they are. +aBc—axry +076 + a2 Y amb Qo A Bae ih ce b2 — a? — c? Observe, that no even power of a expression become the third by changes (a2 at a® &c.) changes sign when a ofsign only? changes sign; how then can the first 3. p+q @tn (rtp), (P+ @-7) &-P) —-(t+qg @qtn (rtp), P-DM @tr” P-?) 4 (4c) (a by bo). ke + 4ag) a + 6) (=e) (2+4ac) (a+b) (C—b) , —(P—4ac) @t 4) (6 +0) Section 12. Multiplication and Division of Single terms. abe a? b a 6? a® b* ab — b eR) | en ee Cc a b ab abxac=abe 3abx% 6ac = 18atbe pg xg@=pqo axax a = at 08 Gs a? Ge. OP Ee Q9(a+ 6b) = 2at+2b6 2a (a@+b+c—d) Qa+2ab+2ac—2ad 3ab(4a—ab+a2) = Wa&b—-3e0B+3abz wee! ] eS 1 Bh [eap ee Bay mension ces Oy iw = |e ha 22 ) an (Ge 3(a —b) = 3a—35b dabe(2ad +3be-Gac) —sebcetoabhe—2w@ebe EXAMPLES OF THE PROCESSES 3a@2(2y=—-GCz2+2—1) = 8arety— 3a 2-+3 a2? — 3u°Z ela Oe e we ey aang ee \etat) yi bts Wo GN ket Ont) ne a ex “by! da” fy | b 2 mt (sap4 2) = Seem, om : n n n n Sa fa ee aa aneee Bas a 2ab 70 a@ 6 | BBN iP A VEN oly Hh ty Pk i uy cet ae | WGN 2 VOR Ss Ua woe oF epee y x Aa By Up Dae ye gate Se) AT Ai Te al Pd 3 Bae | Bae 8b. .2 13 a? a a Gabe Be wiyietgt | 2 Z2abcm 1 9@b Bab wytzev xe B8aebem d4acm vt 1 pi q's p° Me UP a eae 2Qmvt 2m Seta) 8 He mabe b ey(e@ry)? _ y (pa gyt.aly Md puiqi & BOUL i nyy® x (e+ y) fx(p — q)? Pome UCR AE AE CAS Crcluene cae a a a ma—-na ma? a ee een ee ee a I a ee a a a pub 9) 2 8 — 6 g? pee als mn + 8amn am n Rit IN, dt iia: aia 2mn 202 ae 1 B= 6D Li 2 3 ra 1 4 2 3 a2 “CL 348) tee | Ma SECT Os i De ab—ab® a— pPtr2@py pt+2y ab?+ab ab+i ye-yt3savy y-1+3av x x a2 P- Ce yr+ Bary 3s ey —axr*y+ ea 24aPY +30 at xY—Qaa ye 2x4 7° +3ae 7 —2@ay dati Gat+4bi2 bats —_—____. 2a—-4b° (Qa—2b)18 Saqa—H6 OF ARITHMETIC AND ALGEBRA. 63 b Te es l 2 eh a aca 0 Qaxr+y A40an+a b ac—b eer Rr oh a a=- —-—— — ' a 2 ] hee! 1 H x) cue. ee - wr 2 ~ Ltoe xn lit @ Section 13. Separation of Factors, atb=a(1 t= = oS +1)=ab(5+ -) =o(2+2) =7{ae+be) = (0 + = —) Cc Cc G b When any expression A is to be put in a form of which m shall be a factor, then — must be the other factor: for nN A=mx= 4 b2 ve 4ac= B (1 — -) Wttbs - “(= ~4e) ; : x 2 ¥? e@-—-2Qey=a|'va —2 ie “ry —2 )=2 (9 - +) y ( y = ( y y of? y x u xv \ Ly ; 22) =: x (2- 2y\ = oy (5-2) =5(2 igae Bi a aetd=a(ets)=2(a+2) =0(Gr+1) a x 2 eteyr se o(i sod =0(& +- = +1) Pee yf? . ay Lo saty tanya ah - 495-4} a C (a+ 62 —c = (a+ dD) (a+o- rng = (a + b)2 USvesey at \ zr ye oe} eA oa eek Sa ae Rem OL en LU ee CL—y y, : Emde e (> + ao G48 Cc a b S= Le xR | —n —1 nm— 2 m—3n?*+2n n = 1s 7. = —— 2 2 2 3 6 yi ars Ae TB Be or ll i 6 an ag 3 rey, 7 24 } 1 ere e3 Be eseas Mie igh Op paren Pr 2 2 mti 2n+1, —— “RH? wns Saar : 2 = = 9 te =nti-—. CM ete » Osh t= bee ae — Oe x b wo + é e+(bt+exted era #£-a_2at+ be xz—b + ce ee — E OF ARITHMETIC AND ALGEBRA. 1 ji eta’ ato xte avw—b Seis a—bz meas I ] i sets 1—2 LBs sate Ne a xv 1 x 1—z be 67 ] 5 a +F2@+tb+ovt+abtbetea e+(atb+ec)x?+(ab+be +ca)x+tabe (a + b) # —2am% — (a — db) (at b)a-—-a-bx# I 8 = 2 @ an sant Berry a 20 — 2 x We shall find further examples of these operations in the next section. - Section 16. Solution of Equations of the first degree, or Simple Equations. Rule 1. Both sides of an equation may be increased, diminished, multi- plied, or divided by the same quantity. may be removed from one side of the equation to the other, if the sign be changed from + to —, or from — Rule 2. Any term of an equation to +. Let a2 —22=7-— 2 What is the value of x? eo+rer=7+2 22% =9 9 Verification 44 -2=2% 7—4h= 25 Lett v2-—-a=b—# 2 oo = Oe Pr th c= %(a + J) Verification 3 (a+b) -—a= 3 (b — a) b—¢at bd) =4 0% —-aA) Let 32-4 =1227-—9 9—-4=122%—-—32 5=92 5 =o 7 g : 5 21 Verification Lr 4= aaa 5 21 LR, Bok Aa eaten this negative result as showing that Before reading the theory of the ne- the equation should have been written gative sign, the student must consider | 4—-832=9-12e@ which gives @ = : and can be verified arithmetically. ax—-b=cxux—-—d ax—cx=b-—d (a—c)x=b—d b—d Let GC = F 2 EXAMPLES OF THE PROCESSES , : b—d b—-ad—al Le " Verification a pete ipa ABE ts! ARLE Dati Qa a— ec _be-ad a—ec b—d be—ad ec —-- —d= ——— a—c a—e wv ve 4 3 Let = a a ee he les 4 H by Gf r 12 - 12-—- =48 —12- AU 3 4 6x2 +42=48 —32 62% ohx4imeatr Oe we 45 13 48. @v%=.3 aS xr = = oe pee BC ge z x 40 7 Verification — = De bes Ne ‘4 jes 13 4 rea] x Let —- —-=-e-- ar d ab ded ee bed ahd a b d bdxa+adx=abed—abe bdrt+ada+abxe=abcd (od+ad+abjyx =abed abed ee bh aa se ob ; ; x bed r acd Verification a aii Shee p(S war ee ; MMA, PN bed acd, ©. x Se ae eee ee ee et ey ee ee a ie 3 5 6 Multiply by 30, the least common multiple of 3, 5, and 6. 102 —10 — (62 — 24) = 30 —5a@ 107 —10 —6%+.24= 30-527 | 16 9@2= 16°57 =F4 i] ; ee Front | 7 awv—-—A4 4 sa 1OR vas 4 . ay eS SS se —$— ee eS Le Ke Er Se, hr —_— — Verificatio 5 57° P 9 G 07° 95 ( *) 7 Ay 9 i Rue eee a The negative sign in the verification shows that the equation should have been written xr—]1 4—2 x 6 which gives the same value of x. r-a L£-C te eE b Fe f Let OF ARITHMETIC AND ALGEBRA 69 % Multiply both sides by bad ft A dfx—-adf—bf(«—-—c) s=bedf—bdx dfx —-adf—bfxt+bfeo=bedf—bdex dfx—bfxt+bdx=bedft+adf—bfcec , bed f-Padf— obcf — adf—bf+obd Verification “—— ig a fi ee e ee Rae.) Be hina fof 4: bic ERA” bre bat x— a x—c edf—bef—ad+be re ye kee “adf—bftbd pe tg 8 Si Gti Die dfwbf Hod MTEL ge Fn Se a b a Multiply both sides by a 0. aba—b—-abat+@=ab+ber 2 be=at—b-—ab w=5--b-a NOY REE is bea ae bu Gene a Ede, Pie a ax — b ba— ya. 4 a b Days a x a b a b 1 ee | 2 Ot CN ir aes: aay Hig a b a r—-a a—b x—ab Let ah, Staal agEDO 1k ax-@+t+ba—-B@t+r—-ab=ab 2 MN fake 2a bi O° ieee Ea b +L 2 cae akg) agi Be Be ita eT: fee Picteucli a at+o+1 b 4+ 2+ = Pee bite yo nites’ eee ab mao SE ATG r-—a xz —b r—ab Qa+2b0+4+2 PS atl A = = sh were ee b * a ab at+tb+1 : @. extn In the succeeding examples, we give assumes when the value of # 1s sub- only the solution, or value of andthe stituted. value which each side of the equation 70 EXAMPLES OF THE PROCESSES: * x—a ct 2x x—e Let e+ ey WR a he =d-— ‘3 The value of 2 will be found to be @t+tabdt+bert+e ab+a+t+b—1 which will give x a abd—-@b—abt+atbe+e 6b - biabpatoe- i) c+ ox _@+abd+abct+act+bet+ be ab ab(ab+a-+t+ 6 — 1) %@—e @+abd—-abe—aet+cte he a(ab+a+6b—1) and the common value of the first and second sides of the equation is a(bd+d—1)+a(be+e—d) —(c+e) a(ab+ta+6-— 1) a —-2“v7+a Let “+ b Cc 2% — a b _ axt— oe be the value of w, and the value of each side of the equation, are as follows:— a2 at+tet+l1 a be 2-5 Let (a+) (6+ 2) = atb+e-1 2—6 (c+a2)d+an) This appears at first sight to be an equation of the second degree, but it is not So, owing to the occurrence of x* on both sides of the equation. a ab—cd fel odie eae _—_—_ c-a.c—b.a—d.b-—d- (fata) (6+txan =(c+2r7) Eta = c+td-a-—b In verifying equations, and in alge- braical operations generally, remember that addition and subtraction seldom or never take place in denominators, so long as they remain denominators, and only occur when, by the rules for fractional operations, denominators have been incorporated with nume- rators. Also remember that, except for the purpose of incorporating two ex- pressions, the indication of the multi- plication is simpler than the actual result. For instance, we form a better idea of a-++ 6 taken a+ 6 times, which involves one multiplication, and two additions of the most simple character, than of its equal a?-+-2a6-- 6? in- volving two additions and three mul- tiplications. Hence, the most conve- nient plan will be, to let the indications of multiplication remain in denomi- nators, without performing the opera- tions, until, in the course of the process, those denominators become numerators or factors of numerators. Thus, (a + b) (ec + ad) (e+ Ss) (g +h) should be written actad+6bc+bd (e+ f) (¢ + A) When a denominator occurs which will be written several times in the course ofa process, the better-way will be to substitute a single letter for it ? restoring the original denominator only — when the chosen letter comes to appear ina numerator, The following exam- ple is worked at full length in every } ‘ OF ARITHMETIC AND ALGEBRA. 71 respect, containing everything which the student would find it necessary to write :— gon 2 c-a a b =— ee CO le BY, ab bxa-b —-a(tat -—-a)=ab2e- (atb)z ba -BP—axt+@d=abex—ax-bse @—-BP=abre-—2b2 a? — 6 ab—-26 Ht ae be bh — 7.68 abP res De 1 Be. ; (a+ b)x Again, ep - GF-s(i- a -— 6 ab—a—b P ab Multiplications should always, unless where they are more than one and complicated, be performed as in the last line of the above, without arrang- ing the multiplier and multiplicand under each other. When it is evident that the nume- c(h —c) (d — a)-_ GetGe Ay i om dist the sides of the equation become : BPcd—bed - @&c a’ —b?+be In the following examples we shall discuss the cases in which are presented what have been called the creéical values of the solutions, namely, = and 7 which are treated in all algebraical works, and which are here considered only as con- nected with the following theorems, which the student is to verify, each as it occurs, upon the examples given. the forms 0, ia a —b—ab +28 Bb —~aB—-&+abh? + ab—a b _ &b-ab—a + ab? -@b+é6 Letab- 25=P a2 + b2 — a B P — —— P a—- &—~ab+2ab ——— a — b> —a®?b+ 2ab bP a@-ab—@b+2a26 abP @b+h8—~a —~&+ ae? +a2b — 20°) (ist side) a+b oe, ab-a—b ab r aan ab (2nd side) rator of a result is the product of cer- tain factors, that result should be written both with the multiplications indicated and developed. As in the following example :-— ax | jae cd=bx-—-ae chd—Cd—-abe+ac a - B+ be 1, © = 0 indicates an equation of the fom axv+b=cwx + 6, in which a and ¢ are not equal: or an equation which may be reduced to that form. ee 205 r¥ indicates an equation which either has or may be reduced to, the form ax +6=a2x-+c in which J and ¢ are not equal; and which cannot be true for any value of 2. ees ca : : 3. = >» indicates an equation which either has or may be reduced to, the form ax+b=ax + 3b, which is true for all values of 2. alba bei at Nc d a a Cc c Cig “Tet me ty c—-a-—ap The sides of the equation become acep—a@bp+cq—acq “wc({e¢ —a-ap) If cg +.c —:a b. =0,that is, if- ab— @ , the solution becomes x = 0, unless it happen that we have alsoc —a—ap = 0, in which case it takes the form - Let us suppose the first only, and not the second. The equation itself is the same as 1 1\ b Cc p q eapevirsy Buiceae dell we ike ae PH pea Z Miiglte ®) Se fag c c then we Ni 8 at as aeten, which call B. Then the equation is (G-7)etB-2e+s a c 1a joc) |) EXAMPLES OF THE PROCESSES C= which is in the form given in the first - theorem, If at the same time c - a — ap = 0, a-c cate | p= > g]s a hpi eee c Cc a aoe , which call A. The equation isAw+ B=Aa+B, which has the form given in the third theorem. If the latter only be true, the solu- 2 oes b of S » ale the equation itself corresponding to this case is (A meaning as before) q a tion has the form An+2-lL=Anrt which has the form given in the second theorem. The student should now go through the various examples which have been given, and should show that the same suppositions which reduce the value of x to either of the forms 0 w= — 0 will, when applied to the equation itself, reduce it to the form Azxzv+B=C2r+B Equations should also be solved, de- Av+B=Ar+D Av+B=Ax+B ferring all reductions until the result or SO 2g ts OAS tog has been obtained in the form of a PT NENG Re NO AAC AES Aa Mo complex fraction, as follows. The equation Regie + A BPS eta te S os L¢ Pz) ig anit a ac ae a+e c ili cae a ate Mi Asal 2 erie pieapon ith 149 b eile G+2 tec eee okie LCA b ; _ This should now be reduced by mul- patil ; tiplying both the numerator and deno- L= — 5 af : minator by a ¢ (a-+ c), which gives a At! : 12? Lb. Me. Soa neh 3 aera bfabe | (C+ p) (4+ ¢)+ ae ma a—ba Let Se Ta sc: ab OR ree ae c a+ 6 a ead Ny ns mechs Pans el PRG RT OR Sy ees So ee by rsa c Peale Mie b(a + bd) a(a + 6) a a Pa es OF ARITHMETIC AND ALGEBRA. 73 6b b p 1 i Dees moa Sa ere Re er ay GF a 1 1 Gud atact: Oy -5 a abe(a + 6) — Be(a + J) +ab*(a +b) —abcp'+ac @b(atb)+be-—-bc(at bd) One of the things which the student must observe is, that an equation may be, in particular cases, intelligible in some forms and not in others, though the intelligible forms are direct deduc- tions from the unintelligible. For in- stance suppose ee tb <= (1) ex—-aer+bcex=c—-c2x (2) If we suppose c = 0, the equation assumes the unintelligible form r—a 1-2 —— +62 = —— while (2) assumes the form ex—ae= 0, orx =a which is also the result of the solution (3) in the case where c = 0. here nothing to do except with the We have We shall solve one problem in ge- neral terms, that is, by expressing the known quantities by letters whose va- lues are supposed to be given; and shall then proceed to inquire what particular values of the known quantities will give critical or other peculiar solutions, and what is the meaning of the problem in ' the cases thus formed. -It is supposed that the student is already acquainted with the usual me- thod of treating the negative sign, and the following examples are for exercise, not for primary instruction. Problem. There are two pieces of stuff, of 27 and // yards in length; of these the owner sells the same number rg ink a} tk Gt) — 2) 2 ie She, It is usual to write algebraical ex- pressions in some form which shows symmetry, even where cases may occur in which combinations thus introduced following circumstance, that the rules for the solution of an equation give solutions to unintelligible as well as intelligible cases, which the student must connect together by means of the usual explanation of the former cases. At present, let all the examples be looked at, and let the following theo- rem be verified in each ease. When such suppositions are made as to the values of the letters repre- senting known quantities as will make one denominator only equal to nothing, then the same suppositions applied to the solution will give such a value of the unknown quantity as makes the numerator of that denominator equal to nothing. Tt is usual to accustom the ‘student to the solution of various problems producing equations of the first degree : these are of no use whatever in them- selves, but may be made to furnish illustrations of the several algebraical peculiarities of the results. To these we shall therefore proceed. Section 17 Interpretation of the cases of a Problem producing Equations of : the first degree. of yards at pand p’! shillings a yard, and afterwards selling the remainders at gq and q’ shillings a yard, finds the same receipts from both pieces. What was the number of yards first sold of each ? Let x be that number of yards, Then pe +gqtt — 2) is received from the first piece, and ga+aql(l — x) from the second. The equation of these expressions requires that ashould be qi-al piri (gies Gy ~gl@Ma-a =i tpiz gq) are negative. The rules for the nega- tive sign render such cases as manage- able as any others, = px Pa it — V) 74 Let us first suppose that p’— q! = p — q, or that the first and second prices of each stuff exceed each other by the same sum. The solution then takes the form ging 0 and the equation becomes A = p! — g@=p-4 Agt+ql=Anr+d'I which is never true when gq / differs from q'/’, Nevertheless, it may be made to approach as near as we please to the truth by taking a sufficiently great. For it must be remembered, that a +aand x + 8, for instance, are nearly to equality the greater @ is taken. Thus, 1000 + 1 and 1000 + 2 are more nearly equal* than 4 + 1 and 4+ 2. But at first it cannot be sup- posed that xis greater than the least ofZand/’. Consequently, keeping the literal meaning of the problem, it is here impossible. But we will now state another problem, of which the one just given is a particular case, and which leads to the same equation. Problem. A man has two pieces of stuff, of Zand /’ yards in length; he engages to furnish the same number of yards of each at p and 7’ shillings a yard, and having made this bargain, he finds the prices in the market to be qand q! shillings a yard. He makes such new bargains as enable him to ful- fil the first, and leave him without any stuff of either sort. He then finds his receipts (or deficits if he have lost) to be the same from both. What number of yards of each did he first engage to supply? This is the problem in its most ge- neral terms. It meets the case that he may first have engaged to supply more than he has got of one or both; and also that in making up the stipulated quantities, he may be obliged to buy at such a price, that on the whole he will lose instead of gain. Let us first take the case that he has engaged to supply more than he has of either (v yards). Thenhe has to go out and buy x — /and x# — / yards of the two sorts at qg and q’ shillings a ard. This costs him qg(a#@ —J) and q' (« — Ul) shillings, and he then sells his x yards of both sorts for p w and * For a complete explanation of this point the student may refer to the first chapter of the Treatise on the Differential Calculus, which contains nothing more than a student might here read, EXAMPLES OF THE PROCESSES p' x shillings respectively. If the se cond be greater than the first he re~ ceives px —q(«e—ljandpa—q (x - ly shillings for the two, and px—-q«e-bhb=pua-—q(«e-l) by the problem: but if the second be less than the first, he loses q(w@—l—pxandgq' («-—l)-—plex shillings by the two, and gq(a@—-)—-pxr=aq(e«-l)—-pe by the problem. Both these last equa~ tions give the same as in the first case. Let 7 > //, and suppose he can fur- nish the stipulated quantity of the first stock, but not of the second, that is, x is less than /, and greater than 7’. He has then by the problem to dispose of his remainder ? — a2 and to make up the deficiency « — /, at q and q’ shil- lings a yard. Consequently, he re- ceives from the first px +q(/— 2), and from the second he receives p! x and has to pay g! (w@ — /') which leaves him p’x2 — q/(«@—V). The first must be greater than the second, for by the problem he has the same (receipts or deficits) from both, and from the first he clearly receives; therefore he does so from the second. And the equation is pxetqgdl—x=pr-q(«e-l) giving the same value of z as before. Let us take another possible variety of the same question. He does not engage to furnish, but to take, the same number of yards of both, at p and p’ shillings a yard. He then con- cludes such a bargain as rids him of his whole stock at q and q’ shillings a yard, and finds his receipts or deficits the same from both. It is plain that he then first buys 2 yards of both for px and p’a shillings, and sells his {+ a yards-of the first, and /’/ +a yards of the second at g and q/’ shil- lings a yard. If he gain by this, the equation is qi+a)-pxr=aq(’+nx2)—-p'a; if he lose, it is pe-qidt+a=p'e- gq (+x) and both give OF ARITHMETIC AND ALGEBRA. OF gee ph-¢ —-@P-D but these latter only differ in the sign of the numerator, that is, only differ in sign. We shall now return to the general problem, and the case where p/ — q' = 2=20 U=40 p=10 wh REA If we suppose w yards (more than either 20 or 40) to be first bought, the equation is 10% -6 (w@ —20) =8a—4(@ — 40) Or, £42. -- 1200=— 42.-- 160 which cannot be, for the second side always exceeds the first by 40 (shil- lings). But this excess of 40 may evi- dently be made as small a part of the transaction as we please, by supposing x sufficiently great, and if two quan- tities be called nearly equal which have a very small difference when compared with their own magnitude, (which is the usual meaning of nearly equal,) then (Study of Mathematics, p. 41) this problem is said to be solved when x is infinite; that is, may be as nearly l i! p p! I. 50 25 6 6 Ll. 20 20 5 9 2 the Woven at 8 3 8 IV. 64 36 1] 9 The student must remember that the answers to this problem will not al- ways be whole numbers, but that these eases have been so contrived, in order to avoid fractions, and render the point we are now considering the only diffi- culty. I. The negative sign of the answer shows a diametrically opposite mean- ing to that which was supposed in the equation, Wesupposed that the owner began by engaging to furnish a quan- tity of each; the answer shows that the problem cannot be solved in that way, but must be solved by supposing him engaged to buy 175 yards of each, both at 6 shillings; or, if it seems more ‘clear, the problem proposed is not pos- sible, but the corresponding problem which supposes him to begin by buy- ing is possible, and the quantity so -bought must be 175 yards; this is an outlay of 175 x 6, or 1050 for each ; but the stocks of 175 + 50 and 175 + | 25, or 225 and 200 then in hand, sold at 8 and 9 shillings give 1800 and 1800 | not as before p-gd—-Pp-® 75 qi-ql p—q. Say that the lengths of the two pieces are 20 and 40 yards, the first and second prices of the first 10 and 8 shillings, those of the second 6 and 4 shillings, or pa8 q=6 qa 4 solved as we please by taking « suf- ficiently great. If in the general problem we sup- pose g/=q'/, and also p! —q! = p — q, the value of x takes the form ~, and the equation becomes of the form Arvrt+B=Aa+B. Any value may be given tox. If in the preceding in- stance we suppose /’ to be 30 instead of 40, all the rest remaining the same, the equation becomes 42+ 120 =4@ + 120 which is true of all values of x, or any quantity cut off or added to the stocks mentioned in the problem, satisfies the equation. The following are other cases of this problem :— q q’ xv Receipts 8 9 — 175 750 10 6 8 144 10 9) 66 238 9 16 0 576 shillings, so that the receipts for each are 750 shillings. II. Is altogether within the limits of the first supposition. III. Shows that he will have to buy 58 yards of the second to make up the 66 yards which the answer to the pro- blem shows he is engaged to furnish. This costs 58 X 5, or 290 shillings, and the whole 66 at 8 shillings yields 528, giving, as the balance of receipt, 238. Of the first stock he sells 66 yards at 3 shillings, yielding 198 shillings, and the remaining 4 at 10 shillings, yielding 238 shillings in all. IV. Shows that he must not cut off any of either piece in the first bargain, for selling the remainders (which in this ease are the wholes) at 9 and 16 shillings, he just gets the same from both. Let the student reconsider this ques- tion again and again, taking additional examples, and explaining them in the preceding manner. In every subse- quent question (particularly in geo- 76 -EXAMPLES OF THE PROCESSES metry) he must always be on the watch to explain the three forms, namely, ne- Saag a 0 gative quantity, me and x I. When the value of an unknown quantity appears to be negative, look pack at the problem, and consider the negative quantity as unmeaning and unexplained, until it has been shown that the problem requires the unknown quantity to be of the kind diametrically opposite to that which it was at first supposed to be. B08 A 2, Consider the form 7 as unmean- SECTION 18. This subject, though of. very little practical use, will tend to impress on the mind of the student some consi- derations connected with whole num- bers which will make him more expert in common arithmetic. The following theorems and defi- nitions are necessary :— ~ 1. A prime number is one which ad- mits of no divisors, except 1 and itself: such as 7, 29, 31. 2. All italic letters in this section signify whole numbers, and Greek let- ters whole numbers or fractions. 3. All numbers divisible by a are contained in the formula ad. Thus, all numbers divisible by 6 are con- tained in the set 6 xX 1,6 X 2,5 X 3, 6 x 4, &e. All numbers which divided by a leave a remainder c, are contained inab+e. Thus, all the numbers which divided by 13 leave a remainder 4, are con- tained in the set 13 x 1+ 4,13 x 2+ 4,13x 3+ 4, &e. 5. All numbers are either prime numbers, or are made by multiplying prime numbers together. 20.is 2.2.5 641s2.2/2.2./959, 24201s9.2.5.11.11, 2310 is 2.3.5.7.11, andso on. That is, every whole number may be repre- sented by Ts, ROO EK wen XIN. ae where a, 0,c,... are prime numbers, and m, n,q, &c. are whole numbers, prime or not: 6. No number can be resolved into prime fuctors (the process of the last) in two different ways. Thus, if adc = def, it is impossible that all the six, a b, &e., can be prime. 7. If a divide 6, the prime factors of a are all among the prime factors of 6. Thus, 360 is 2%,3°,5, and all the divi- ing until it is shown that the greater the value given to the unknown quan- tity, the more nearly is a solution pro- duced to the problem; then, and not before, use the abbreviated form of speech that ihe unknown quantity is infinite. 0%. 3. When md the result of an equa- tion of the first degree, let it be clearly ascertained that any value of the un- known quantity is a solution of the problem. What it means as to an equation of the second degree, we shall afterwards explain, On Equations which are required to be solved in whole Numbers. sors of 360 are represented by a case of one or other of the sets 2™ 39,5, 2,5, 39,5, 2.3m, 9m gn 5 where mis not > 3 7 is not > 2, 8. Ifa number represented by its prime factors, be a™ 6. cP, the number of di- visors does not at all depend upon a, 6, and c, but entirely upon m, , and p. Thus, 600 being 2°.5?.3! has the same number of divisors as 360, or 23.375}, The number of divisors (unity and the number itself included) is (m+ 1) (n+ 1) (p+ 1). Thus, 360 and 600 both have (3 + 1) (2 + 1) (1 +1) or 24 divisors. We exhibit those of 360. 1,°3, 37) 2,973, 2, 33,°o", Sts leas 28/2813, S253", 5, O70) Oy ae, eee 522 BF G22 248 aS eee Oo. 278, 8.08 aa Give the reason why the number of divisors is here (3 + 1) (2+ 1)(1 +1), and try to give a general demonstration of the theorem. It is required to solve the equation 11 xz2-+-7y = 108 in whole numbers, or to find all the values of x and y? Or it is required to divide 108 into two numbers, one a multiple of 11, the other of 7, in as: many different ways as pos- sible. The process is as follows :— Since y is a whole number, or } (168 — 1lz), or15 —-a +1(°3 — 4 a), of which 15 — zis a whole number; it follows that + (3 — 42) is a whole number. Let it be #; then3 —-4a”= 7t, ore = —i+1(3 —3 2#).- This algebraic fraction is to be a whole num- ber, let it be z’, then 3 — 3¢ = 4 7, or #=1-7 — 27’, so that 3 ¢/ must be a whole number. Let 2” be this whole number; then#’=3 7" ¢=1-37#/=— Mol—4"s; w= —1+4it +4 S-_ 3+ 120) = 71 = 1 . OF ARITHMETIC y=15—7"+14+3(3 — 28 ¢" +4) =17— 11%) EL ee Pye ea TE ETS’ = 77th 10S This result being independent of ¢”, it would seem that we have thus an in- finite number of answers. And so we have if we consider algebraical answers, that is, positive or negative whole num- bers; but if we restrict ourselves to arithmetical whole numbers, that is, to positive algebraical answers, we must so assume ¢” that 7 7” is greater than 1, and 112” less than 17. The only value of ¢’’ which satisfies both conditions is 1, which gives 2 = 6, y = 6, or 11.6+ 7.6 = 108, which -is the only arith- metical answer. As these quéstions are entirely for exercise in numbers, we shall give no rule, but only a method. It is required to find a set of numbers, which being divided by 4, 5, and 6, give remainders 1, 2, and 3. Let us consider the two first conditions: let x and ybe the quotients (rejecting remainders) of this number when divided by 4 and 5; hence, since the remainders are to be land 2, we must have 42+ 1, and 5y +2, both equal to the required number. Consequently 4%¢+1=i5y+2,0or4% -5y=1 must be solved in whole numbers. We might find this as follows : we want to find a multiple of four which exceeds a multiple of five by 1. A case is evi- dently found where 2 = 4,y = 3; add to these any multiples of 5 and 4, such as 5f and 4 ¢t, and we have v= 4+ 5t =3 + 4% A445, — 5 (3-4) =i) . But the preceding, though it shows that these forms satisfy the conditions, does not show that they are the only forms which do so. To show this, pro- ceed as before: we have rv =y +4 (y+ 1), therefore y + 1 must be of the form 4¢or y of the form 47 — 1. The preceding gave 4 ¢+3, which does not appear at first to be the same; but it is so in reality: for though 4t — lis not the sameas 4¢ -+ 3, for any one value of ¢, yet it is plain that lless than one multiple of 4 is the same as 3 more than another. And the value of ¢ may be any whole num- ber. Taking y = 4¢- 3, then the inumber 547 + 2 is 20é@°+ 17, which formula contains all such numbers as, AND ALGEBRA. 77 being divided by 4 and 5, will leave remainders 1 and 2. But this divided by 6 is to leave a remainder 3, by the last clause of the problem. Divide by 6 algebraically, giving 3#4+2 + 3(2¢+5), but 3¢ +2 is awhole num- ber, therefore the remainder 3 must come from 2 ¢ + 5, which must there- fore be of the form 6 #/ + 3. But 2¢+ 5= 62 +:3 gives ¢ = 3¢/ — 1, and 20% +17 = 602 — 3, which is the form required. For example, let /=1, then 57 satisfies the conditions; let t/=2, then 117 also satisfies them; and soon. The student may make abun- dance of examples for himself, the test of correctness being obvious. Theorem. Show thataxw-+ by=e eannot be solved in whole numbers if either two of the three, a, 0, ande, have a common measure which the third has not. We shall now suppose it required to solve the equation ae -- y? — 2 in whole numbers. Always in such a case endeavour to write the equation so that both sides may be reducible to a pair of factors. In the present instance write gS Zh 7%, or ee= (2 -yety) 1. Suppose neither of the three to be given. Assume y L v , » 2 v 2 v Assume™ any even number for a, and let 2 ube any divisor of 7. Or assume a any odd number, and v any divisor of it. For example, Jet 7 = 9,v = 3, then 2°= 15jcy ='12; and. 9° Pela = 81+ 144 = 225 = 15%. This also con- tains the case where 2 is given, for it must then be assumed as given. 2, If z be given, the preceding equa- tions give Say =F ex nen. 2 AN bi fe) Bie pre De les a ey Write - for v, which gives (7 > 2) 2 2 2mnz -p mn —n Ps 4 ry © m* +- n* m + PA o * Any proof that may be wanting is left for the student. 78 Prove 1. that if m and” be whole numbers with a common measure, the preceding can be reduced, and other numbers, which have no common mea- sure, substituted for them without al- tering gory. 2. That in the latter case mnand m? + n? cannot have a common measure. Hence, prove that when @ and y are whole numbers, the problem is impos- sible except when 2 or one of its fac- tors is the sum of two unequal squares. Thus, when zg = 5 or 4 + 1, letm?= 4 ges and @ = 4°7°= 8)" Bat wher z= 11, the problem is impossible. When z= 10, or 9 + 1, then m= 3 nm=lgivesv@=6y='8. But 5, afac- for of 10,is4+l,andm=2n”=1 gives © = 8 y = 6, which is only a reversal of the former. Show the following, I. Ifm be greater than 2, and both be integers, then C=mM—-nv y= tnn L£= 78 +78 satisfies the preceding problem. II. The square of an even number is divisible by four, and that of an odd number must leave a remainder one. III. A prime number divided by 6, must have 2mn+n? = P12 = . LL woe (Ja And from a preceding part show that the following are solutions, 7 = m? —n?, y=aQlmnt na 2= 7h mn +e: For instance, let m = 3,” = 2, then EXAMPLES OF THE PROCESSES a remainder 1 or 5. IV. One more than a square may be a prime number, but one more than a cube cannot. Problem. To find two numbers, of which the.sum of the squares added to” the product gives a square, or to solve. Chey hy =e | show that, if vy, and 2, satisfy this equation, any of the same multiples of the three also satisfy it; and hence prove that any real fractional solution gives a solution in whole numbers. Show that, in this problem, either of the following equations is a conse- quence of the other. e+y =v(r+y) e-y = From these deduce Pa: @ mod ear ue 1 @ 2-v' v — Faw ig Show that » must be greater than 1, “ead ———) and less than 2, or that ifv = m must be less than m. Substitute this value of v, and thus obtain the results m? +m n+ n? . m? — n* x=5, y=16,2 = 19. and find other instances. Show that if 2 be greater than m, then Verify this, c=n—-m y=2mn+n z=m+mn-+ x? satisfy 1. By the theory of the negative sign applied to the preceding. Apply a similar process to av+tbay+t+y* = 2’, a, b, and c, being given whole numbers. And, finally, apply the same process to axv+bay+tecy?= ez Problem. Required two numbers, of which the sum of the squares shall be a given number of times the sum. Let m be the given number of times, so that Bo pie am (ey) Assume = — y, which gives by substitution _ma(p+q) _ mp (p--q) Gene gah ck eee Var ie hae Sager a RIC Raa pore ai a If possible, take py and q so that e—-xry + yi = 2 2. By an independent process simi- lar to the above. ae@tbaey + cy? = cz m = p*-t- q*, or m= c (p?-+ q?), where cisa whole number. Then y=cqap+g e«=cp(p+q for instance, if m = 25 = 5 (4+ 1) x= 15,y = 30. Problem. Required triangular num- bers which are also squares. A tri- angular number is any number which results from making # a whole number ini} a(x+ 1). Show that this must give a whole number. Wehave 4e(e#+1=y¥ Assume omy thena +1=— 2 hi y n m m 2 2 n? m — 272 ~ m2 — 2n2 or oy = OF ARITHMETIC AND ALGEBRA. If we can now get m and m two whole numbers such that m? — 2n? =1, we haveasolution iny= mn, x = 2n?. Let one set of values of m? — 2? = 1 be found, say p and q, we shall show how by this one to find another. Assume m-+ 1 aun whence m-1=2 4% 2pq Pt. 2g giving 7 eee Ty ine roar let » and q be the instances which SECTION 19. Let [z,p] be the abbreviation of the product of all numbers between # and m, both inclusive. Thus [4,9] means 4xX5X6X7X8X9 Let I'm stand for the product of all numbers from lup to ” exclusive: thus I 8 means 7X6X5X4X3X2X1 Now show the following: C(n7 + m+ 1) C598 te | rie [pn +mn]) Tamatm+ i [m+1,1] ° DnxYr(m+2) cpa Mkiaud [w,n + q] whole number, if p be greater than gq, and m + p greaterthann + q. Take instances, and show this proposition: for example, 6.7.8.9.10.11 Peas Oss, O and from the instances endeavour to collect a general proof. If there be 2 counters marked C, C, ig Chine es G the number of different ways in which counters can be drawn, one after the other, counting every two orders, how- ever slightly they differ, as different, is [z,% —p +1]. These are permu- tations of p out of. But the number of different ways in which p can be taken out of” at once, is [z,2—p+1] divided by[1,m”]. These are combina- tions or selections of p out of x. Question. How many different hands can be held at the game of whist, or how many combinations are there of 13 out of 52? must be a and if p be less than m, n? 79 satisfy the preceding : hence it follows that p?> —-2q?=1, and n=2pQ, m = p?-+ 2 q*, which give m? —27?= D2 g? Fi 4.9%. = (Ns. 42 Ot Suk anda = 8p? q y = 2pq (p* +29’) For instance, if p = 3 q = 2, and ? —2q?= 1, which gives from the first method w =8 y = 6, and we have 4.8.9 =6 X 6: and from the second = 288 y= 204, and §$°288,289= 204.204. Observe, that we are not sure of thus getting all solutions; for it is not ne- cessary that mm? — 2 »* should be 1, it is sufficient that it divide mm and 272. Permutations and Combinations. Answer. [52,40] [1,13] How many different choices of 5 may be made out of twelve persons ? Answer. TRE IO.9.8 FULT .8 aes, OF 1. 2. 3.4.5 [ee ae ge this method of abbreviation must be learned by practice: the 3.4 in the denominator is equivalent to the 12 in the numerator, and the 2.5 in the de- nominator tothe 10 in the numerator. In how many orders may four be drawn out of 20? Ans. 20.19.18.17, or 116280. Question. There are four boxes, con- taining a, 6,c, and d balls. In how many ways may 4 balls be drawn, one from each? Ans. From every ball in the first arises b methods of drawing from the first two; there are c times as many ways of drawing from the first three ; for every way of drawing from the first two may be followed by any of the c ways of drawing from the third, There- fore adc is the number of possible drawings from the first three; and by similar reasoning, a bcd is the num- ber of drawings from all four. In the preceding question, how many ways are there of drawing three from three of the four boxes ? Ans. The first thing to be done is the selection of the three boxes to be drawn from: this may be done in (4.3.2) + (1.2.3) or four different ways (an abbre- viation of this sort might be used: in so many ways as three can be taken, in so many ways can one be left; that is, the number required must be four.) or 635,013,559,600 or 792 80 Calling A BC and D the four boxes, the selections may be ABC, ABD, ACD, ores OD; and the drawings from each set may be done in abc, abd, acd, orbcd ways; whence the total number is abc+abdtacd-4 bed. There are 3 boxes, with four, five, Balis are taken out of A (4) B (5) C (6) 0 y) 2 2 0 2 2 2 0 1 1 y) 1 2 1 2 1 1 If there be » counters, all of a differ- ent mark, the total number of different orders in which they can be arranged 19° .[ 7,1}... W bat Case is this OL ‘a preceding theorem? If there be 2, counters marked C,, 2, marked C,, and SECTION 20. The following points contain the summary of the theory of expressions of the first and second degrees, which is one of the most important parts of algebra, and without which the mere solution of equations is of little use. First degree. Let mx +n be the expression of the first degree with re- _ spect to a Let Rbe its root, or the value of which makes a+ 2 =0; then i n 1. R the root is — —, m 2maetn =m(ae — R) for ali values of x. 3. When wis greater than R, m a+ m and mhave the same signs; when wis less than R, m a+ mand m have different signs. * Study of Mathematics, page 49, EXAMPLES OF THE PROCESSES and six counters. In how many ways may 4 balls be drawn, not taking more than 2 from either? Firstly, consider how many ways 4 can be composed out of three numbers not greater than 2 (@ being ineluded); this gives only 0 + 2+ 2 andl Se a, } This gives as follows, since there are three different ways of varying the cases :— | Number of ways of taking them, | 5.4 x bie or 150 Ja 2, & oe) lor) Oo or 90 Be 5.4 x — or 60 2 6.5 or 300 2 4 xX 5 X x 6 or 240 x -3 es ra. x 6 or 180 In all 1020 nm, marked C,, then the total number of different arrangements is [ao ime ar [7,51] [rs1] [m,51] Deduce this from what precedes. Ty- i) . ne o On Expressions of the first and second Degrees. Example. To verify these rules in the Gasomie @ 2 yeandi eee ‘ o 5) if ines wv R=>5 ma2n=— 7 2 Cine giee 2D ple = 2Gr =a 27 ‘ 7)= “f Let x be greater than 5 say = ie then 2x2 —7= 383 and 1s of the same signas 2. Let «be less than - say * — 1: then 22% —7 = — 9, and is of a different sign from 2. Again, let w be positive, but less than > say = 3, then 272-7 = —1, and differs im sign from 2, | 7 EEE o———————————ee—w—eeeeee eee eee ere eee ee eee ee ee ee eer e_ WOO Eo ——————————oO7V——— ee OOO OF ARiTHMETIC AND ALGEBRA. Let x be greater than — 15 1, Let & a oy Lee, # — i @ e 6 3d. Leu. & ra) Let w be less than — re 1, Let ex = — — — figs =, 9, Let cm - — i. 6 Verify these assertions upon other expressions, such as x + 1, $4 —- 3, c. Show that the root of m#+ n+ m'a-+n must lie between the roots of mx + nand ma + n’, by several instances, and by algebraical reasoning. Show also that the root of the first is greater than, equal to, or less than, the average of the roots of the second and third, according as mm’? + n! m? is greater than, equal to, or less than m m/ (2 + n’) if mm! (m + m’) be positive : or according as 2 m!2 + n! m? is less than, equal to, or greater than, mm! (2 + n'), if mm! (m + m’) be negative. mae+nxt+r sit) Ng ages Ringe ma—-nat+r —mer+tnxe-T General properites. 1. 1f62 ~4dac be positive, there are two different roots: if b> —~ 4ac=0, these two roots are both the same: if 62 — 4 ac be nega- sen te eae BM OF 24 © either of these substituted for ~ makes 81 Drak aes 1 Tae Me ae oe 3 ( 2 = L— — 2 ae 1 1 3 Sop eens eal eet m2 Reem tat ona 5 10 2 ¥ 1 — 5 ee" er eo © t) 17 eo Ce ——= 10 e &€ @ 1 1 3 6 i ee / (et — 5 TOs 2 ye Ss: 2 Ai 20) Second degree. Let aa? + ba+ec be the given expression of the second degree, where a, 6, and c may be seve- rally positive, negative, or nothing: and let m and 7 be the absolute numbers ina 6 andc, without signs; so that if a and 6 be positive, and ¢ negative, we mean that a is +m, bis + n, andcis — 1. In making a summary of the pro- perties of the expression above given which are most useful, we distinguish 1, those which. are common to all cases; 2, those which distinguish the eight following cases from each other: me+nx—r —-— mx —-nx+ at ma—-nwvw—r st RT gio th tive, there are no possible roots what- soever. 2. The roots of the expression aa? + 6 x + c, when they exist (call them Ri and Rg) are i ee A Ae eC av@+tbatc= 0. 3. Between the roots and the co-efficients the following equations exist :— Ke Ke! = -- c Ri R, — ia 4. The following equation is true for every value of (when there are roots). ax+ba+ec=a(e — Ri) (@ — Ry). 82 EXAMPLES OF THE PROCESSES 5. When the two roots are equal, that is, when J2 — 4a@c¢ = 0; then B=R=- avwz+bx+cz= a (a — Ri)? : anda a? +b + ¢isthen a perfect square with respect to a, giving Vow +ba+ ache wl Va(e+ a 6. The following equation is always true, and shows (see 1, preceding), that (aa? + ba + c)a is the sum of two positive quantities when 6° — 4ac is negative. (QQax+b2%?+4ac— 8 | 4a | 7.aa% + ba + cand a never differ in sign, except when the two roots of the | former are possible and different, and x is taken so as to lie between them. 8. When there are no roots, the least nwmerical value of aa?+ bu + ¢- happens when avwz+bate= é 2axr+b= , re2=>— — | The preceding properties occur so bra, and particularly in geometry, that : continually in all applications of alge- thestudent should know them perfectly. — Example 1. What are the properties of —-2a?+ 427+ 3 aa — 2 6=+4 cam+3 1. 6 — 4ac= 40; there are two different roots. 2. These roots are Alige yes re 1 Ri = —— =.= 5 10 (ep) ey Wee s , site R, x) ionadie wd All = 1 lig (pos.) oer 2 3. Ri + R= 2 RR =—5 4, -227°+42+3 = —2(x@ — R) («4 — R,) ass 9 (@ mi bled V10) (2 dm M10) . Does not apply to this case. Qn —4 4)? — 4 6. ~2et4e4¢3=0—-*2tO—” — —2(xr —'1)* + 5. 7. —22°+ 424+ 3 is negative for every value of 2, except those which lie between 1—3%10 or —1°5811389 nearly ee a 1G one At 68s 19S Kuen ae in which cases it is positive. Show that it is positive when v%=1, 1°5, or 2°5. 8. Does not apply. 4 Example 2. 3a”? - 122+ 12. § 32° —12@%+4+12=. 3(4@— 2)? | 7. The expression is always positive, — Cae be ae CFs except only when w = 2 (0 is neither 1, 6 —4ac=0; there are two 4 nor -—), equal roots. Example3, 22°+ a+ 1 2. These roots are both = 2. a=2 b= 1 c= 4 . OF ARITHMETIC AND ALGEBRA. 83 ae b2—dac= —7; there are no 8. Its least value is f which is when roots, 1 Ss yl Mi kee 6... 227 -b ade Fos 8 When the co-efficients are fractional, 7. This expression is always positive. Teduce the whole to a common deno- minator ; for instance, reduce 2 — ee ee. ig Ce ee 9 2 3 4 12 then from the properties of the nume- 4, In + ma® -nxv+r) + —-+ Per: those of the fraction may be ob- and — ma@+tnex—rj —+- Particular properties. The pairs the roots, tf any, are both positive. bracketted together in page 81, present 5. In+ met ne—rl +t — ‘no distinction whatever except differ- and —- ma*-nxa2+ry—-—+ ence of sign. Each one is the other : ee ! There must be two roots of different of its pair with all the signs changed. a mp the numerically greater root 1. When a and c have the same “J&"S> sign, the roots may be impossible, and being negative. we have ig Rae gles meee 6?— 4 ac is less than 62 and— mav@+nxaxt+ry) —- ++ VR —4ac >» » » ©,numerically,. There must be two roots of different 2. Whenaand chave different signs, signs, the numerically greater being the roots cannot be impossible, and we PO0sitive. ve have These may all be contained in one ie 4 : Wen Es rule, as follows: — there may be as Sere Ae greater than many positiveroots as there are changes VB — dac + » 0,numeri- of sign from term to term of the ex- cally. pression, and as many negative roots 3. Intme2+naet "| ree a as there are continuations of sign ; and d . according as the first term and the SRNLN Fees 00, Ee Qe 70) Ee second present change or continuation, the particular distinction is, that the the greater root, numerically, is posi- roots, 2f any, are both negative. tive or negative.. i Giese ane —4ac Show. that S i A gee bo ee and from hence that R; = actin Rela R, ee mod tsa b+ Vi — dac b— VB —4ac and also that if a be supposed to vary, Why cannot this be ‘made clear zz becoming smaller and smaller, both cases, when the first forms are used? Also show from the preceding R, continually approaches to - 7 that when ais very small, ; bogie SEs am 2 R, increases without limit. Vi —4ac =b- = nearly. If the roots ofaxz +ba+ec beR, and Ra, those of cw? + ba + a are and . Show this by actually forming the roots. Show that a change of sign in & ‘What are those parts ? changes only the sign of the roots, and Ans. The roots of the expression not their numerical magnitude, a2 —-2p x + q’, or the solutions of the Example, A number 2 pis divided equation into two parts, whose product is q’. a—2pxex+ Hee 84 namely, p + N 2 — ¢ Prove from this that it is impossible the product of the two parts into which any number is divided should exceed the square of the half: for instance, that no two numbers or fractions which added together make 10, can have a product exceeding 25. EXAMPLES OF THE PROCESSES and p — “p> — q Example. What are the solutions 0 (x — a) (w@ — b) = c? Aye b+V(a — by? + 4c. es What are the conditions of possibility of this equation? ing, which is possible ? (7 — 4) (wt — 1) = 12 (@-—1) (@® — 2) = — 100. Example. When is the following ex- What do the roots become when c = 0? How does this appear from the pression possible? equation itself? VG —4ac) P+ @bd—Aaexrt+d—dAaf.... (1) The roots of the expression where square root is shown are Qae—bd+t Vial(cd +ae—=bde) + (B—4ac) ft . GF - AaC Of the two follow- c j For whai values of x will the following equation allow of being solved by pos- sible values of y? ay t+baey +cat + dy tert y=) Show by solving the equation on the supposition that y is to be found, that this question reduces itself to the pre- ceding. Show that the roots of aa + 264+ por Aare What are the roots of Cl Fa) 2? — 2 (1 + Ans. 1 and For what values of a is the second root negative? Find this out from the expression, without looking at the root ; and from the root without looking at the expression. ; dns. When a lies between — : and —J. Question. If the difference of the two rootsofazv? + ba+c=0 be D, what are the roots, and what equation must exist between a, 0, c, and D? b—aD?=4aec. Prove this in two different ways. Question. There are two numbers whose sum is p, and the sum of their squares is g: what are the numbers? —b+ Vie ac. a As this form often occurs it should be remembered. for the roots ofaz? + 62+ c¢ = 0, and explain the difference. 25) Se tee Oe i+ 34a : We i Ans. p sat eles) Po eh Show that w + - cannot be less than 2, or that ] © ee 2a (@ el) has no possible roots. If the sum of two numbers be p, and the sum of their cubes g, then those two numbers are the roots of the ex- pression 4 a— pet EE 3p If the roots ofa a + ba +chbe Ri and R,, what is the expression whose roots are Ri + k& and Ry + Rk? Ans. av? + (b6-—-2akjx+akt- bk+¢ Compare it with that OF ARITHMETIC AND ALGEBRA. What is the expression where roots are m Ri and m Ro?. Ans. av®+mbset+ me. verify and explain both cases. Show that » and v® are rational when a is a number or fraction added to its square, and that only one square root of v” will satisfy the condition. Find also the }— Oy 4a 85 What number is that which exceeds its square root by a? Let wv be the square root, and v* the number; then a 1+4a a Lt ta Miiaial 2 number, which added to its square root is equal fo a, and show that the square root of v* which does not satisfy one question, satisfies the other. What are the roots of a v* + ba +c? pone) J akon Ve —4ac 2a Point out the cases in which two roots only are possible, and in which all are possible or none possible. Gt dha) Mot ee Ve = 440 2@ Show that there cannot be one or three possible, and the others or other impossible. What are the roots of aa" + ba" + c? n ga a a aE a Ans. hol rl + Vb = 4ac 2a which may be taken with either sign if nm be even. We do not here enter on the impossible roots. What two numbers are there whose sum is m times their difference, and whose product is 2 times their sum ? If wand y be the numbers, then we have the equations aty=m(e£-— y) Cy eH moar) m— From the first, y = Sergi ed which substituted in the second, gives ——— 2mn 2 d thence y = ——-—. biome? vain he I What numbers are there, the sum of e+y=am(et y) =— From the second, ti ze RT) Al en a 8 : 2a whose squares is m times their product, and the difference of whose squares is nm times their product. Examine the equations and prove that these two con- ditions are contradictory unless m? — n2 = 4, Thence prove that they are never true when m is a whole number, unless m =2 m= 0, and never true when nis a whole number. Show, as in page 77, that when 1 1 m=zk+ — n=k—-, % R k any values of x and y which satisfy the first of the equations also satisfy the second. Solve the equations Vryan(an- y) i— 1; ‘ar, Gola _ which, substituted in the first, gives, after reduction, 9(n? +1)a®*= 2mn (n+ 1/2. Ans. Eitherzwz=0 - y= 05 n+l nm— 1 =m 2 —-— y= mn——— ora =m aoa y ay Solve the equations u- Y= MLYses (A) LTe-younuvy..- «Af{B). First deduce @ + y = = ae eh a a 86 EXAMPLES OF THE PROCESSES Substitute y obtained from this in both of the first equations, and show that. they produce different results, namely, n? (2 + m) a? + mn(2—m) x2 - m= nxw+t(2Q—-mnx—m=0 a Dog see ram @0)) the roots of either of these are values of 2, which with y = ~ — x, solve the equation ; namely, (D} 6°. vis (E) rs) The truth is, that by the artifice of division, which produced;the equation (C), we have obtained equations of the second degree ; whereas, had we simply substituted in (A) the value of y from (B), namely, iC chee ee +17 we should, after reduction, have ob- tained an equation of the fourth degree, which may have four possible roots. We must here remind the student that there is a degree of connexion be- tween algebraical equations, more than is actually and logically contained in the forms of speech by which they are connected, Let us take the following assertions : (A) All right angles are equal. (B) P and Q are right angles. (C) P and Q are equal angles. If (A) and (B) be both true, (C) must be true; but if (A) and (C) be both true, it does not necessarily follow that (B) must be true, as will easily be seen. But take for (A) (B) and (C) three algebraical. equations, of which (C) necessarily follows, or is true, when (A) and (B) are true; for instance, (B) B@—y=5 (C) C7 OP = BD, of which (C) must be true when (A) and (B) are true. It follows that when (A) and (C) are true, either (B) is true, or (B) is one of ¢wo equations %n this case (it might be three, four, &c., in problems of a higher degree) one of which must betrue. Assume equations (A) and (C) or Gry=7 9 ae" = 35. If we divide the second by the first, we have x — y = 5, which it appears at first’ must follow absolutely; and this is true, finite numbers only being con- sidered, But we have (Study of Ma- =m (2 = m) tm N12 + mt 2” (2 +m) ered at) . ete Vn (2—my-amn. 2n thematics, p. 41) to consider the possi- bility of an 2nfinite solution, or of this problem being one particular case of a general problem, which admits of more solutions than one, in general, but of which solutions one or other increases without limit, as the general problem is made to approach to the particular case in question. We have seen that a problem of the second degree must generally have two solutions, but this seems to have only one; for since equals divided by equals must give equals, it follows that fa+ y =,7, and a — y? = 35, we must have 7 — y = 5, and wx + y = 7, together with C—y= 6 give ¢=6 y= 1 and nothing else whatsoever. The question is, what is become of the second solu- tion which the general problem aa + by=m px#—gqy =nwill be found to have? To solve this question, we must first see whether we really have only one solution. Instead of dividing the second equation by the first, sub- stitute in the second the value of y from the,first, which gives vw? —(7 — 2) = 35, an equation of the second degree at first sight, and therefore with two roots ornone. But on looking further, we see that this equation is no more of the ~ second degree, than x = 1 is of the hundredth degree (being # + a” = 1 + 2’), as the terms of the second degree destroy each other, and leave 4e2—49=35 w=6. But let us now alter the problem by proposing : ety=7 (+ )2* —y? = 35, We here present a problem which may be made as near as we please to the former, by taking & sufficiently small, and which absolutely becomes the former, when k= 0. Now substi- tute as before, and we have OF ARITHMETIC AND ALGEBRA. 87 (1 +k) w — (7 — av)? = 35, or kRa®?+ 14% = 84 ~ 14+ 196 + 336k _—————- — 2R ~ ro 7 t+ 49 + 84k R (p. 83) = V49 + 84k+47 If we now suppose & to diminish continually, the first root approaches continually to while the second is always negative, and has a denominator which dimi- nishes without limit; that is, the root increases numerically without limit. When & is small, the first expression for the second root shows that it is R ease ae fe 2 ; 1 so.that if k were? the root would be nearly — 14000. Let us suppose & to be very small, and let the great and negative value of z be taken. Then 2 +y = 7 shows that y or 7 — & is a little greater and positive. We now ask, what is the substitute for the equation 7 — y = 4, which is necessarily true in the first problem? Is it nearly true in the se- cond problem? To investigate this, _ take the second equation in the form (f*'— y*) + Ro’ = 35, divide the three terms by the equals 2% + y, 7, and 7, which gives b nearly ; —~7 —~ Mag+ 84k Rk 84 Ob Pere fee SES EES BIE W749 + 84k —7 2 cm yr afi = 6. 7 This equation will be true, as might be proved by actual substitution: but it will not be true, because 7 — y =5 nearly, but because x — y is a negative quantity, and } k a is a positive quan- tity greater by 5. And though may be small, w is so large that & x” is con- siderable. With respect to % = 6, which will give y = 1 very nearly, which is one solution of the second problem when & is small, we, may say that w—y+rhkar=5 means wv — y = 5 very nearly, be- cause the term rejected is only about 36 sevenths of &, and is small: but we may not call the latter equation nearly true of the larger root. Question. If aw +ba+ ec, and a x2 +b'a +c! have a root in com- mon, what equation must exist between G; Dice dy On Gk Let P be the common root, and let Q be the other root of the first, and Q’ the other root of the second. Then we have cl PQs Fees (4) A Be ae on Ch) ! P+Q a gar MARE) From these equations deduce the following: Q’— @ = pan A aa from which deduce SS pS Mittens Now, from (1) and (3) deduce a@’+bQ+e =0, and thence (from the preceding) c(ba — Va)?+ b(bd —W a) (ad —dc) +alacd — @ co = 0, which show to be the same as (ac —alcP? = (cb' —db) (bad - Ua). eo « (5). 88 Now deduce the same as follows :— since P is a root of both, we have aeP + 6 P+c¢=0 agaP+heP+ca= Hence, deduce (acl —a'c) P? —(cb'—clb) P= 0 and also (ba! - Ua) P— (acd —a'ec) = 0, and from the two last deduce (5). Question. Given the values of a and c; required the method of finding such values of 5 as will make the roots of ax? +bxa-+¢c rational. Show, from the method explained in page 77, that 6? — 4acis asquare when Cc b=am+ —, m m being any number or fraction what- soever. Show that the roots in this Case are iC —m and = ===2 am " Question. In how many different ways can whole and positive rational roots be given to the expression 32° + ba + 36? Show, from the preceding, that m must be either —1, -— 2, —3, —4, EXAMPLES OF THE PROCESSES — 6, or — 12, and that the values of 5 - are — 39, —24, —21, — 21, — 24 © and — 39. Explain the recurrence of the values of 5. Miscellaneous questions. Deduce from the general equation, the roots of azv’+cz=0, and of aa+ br= 0. On what does it depend whether the roots of the first are possible or not? Show that when 6 is very small com- pared with a and c¢, the roots (if any, on what does this depend?) are one positive and the other negative, but numerically very nearly equal. Show that when c is very small compared with aand 8, that one of the roots must be very small, but not the other, neces- sarily. What are the necessary con- ditions that both the roots must be very small? If ¢ and abe equal, what re- lation must exist between the roots? Show that if the sui of the squares of the roots be unity, we must have 2 — 2ac=a’*, In what relation do the roots ofa 27+ mba-+ m‘c stand to those of aw? + ba+c. Show from the method already given, that if aa®+ bx2+ cand a a?+bha+dc¢ have a common roof, (atpa)2’+ 6+pbh')a+ct+pe! has the same for all values of p. SECTION 21, On Equations of the first degree, with more unknown guaniilies than one. As few cases of these will occur in the future reading of the student, which present any very complicated opera- tions, we shall here only describe the best method of proceeding when only one of the unknown quantities (three in number) is wanted. Suppose the equations to be 22%+r3y+42z= 20 32—-2y—-2z2= 4 A Oy a ae ae A Suppose the value of 2 is required. Multiply the second equation by p, and the third by q, then add the three to- Bees and suppose p and gq {o be such that 2+3p +49 See) 5 me Show that we must then have AO rap tae Be = 0 == 0K that from the preceding equations, se fe whence 0 =—q=--—, z= 0. P 5 q 5 Again, suppose the value of 2 is wanted in axt+tby—cz=1 bxat+tcy—az=1 cx+ay—bz=1 Show that if be ot ag= ec+tap+bq=0 SS Mart Show 0 then Ser awn wer opt and that B— ac . @—adb E Bie Oc ~ @— be a a+h+eé—ab—be—ca a+6+ 8 — 3abc¢ _ Determine the values of w and y by a similar method, OF ARITHMETIC AND ALGEBRA. 89 Secrion 22. On Exponents. Explain the following expressions : 1 1 2 2 a Ere gee Ps sagt Nee Dot mes. Piso Grn eee) 1G m\ @ 3 2 Car)e, Yew 1)? Is a” Ne x a i ccaleg ee x 5 ae ae ean a mer a” = m— in a” oo ae ae wen ean x oe» ae agticae pl Bere Fie ee BAY an ake oe TE vee ) — pas: 3 gre F = aaa ( xr nm eee a n 2 3 - 4 = 3 -4 2 CK eH Ke = @ LX UL —2e =2 What is the value of » in the following equations ? P 6 4 - be rks ae 12 Qe Maas FN oR Ns Die) 2 Re ee : Reduce the following expressions to fractional exponents and find the results : Nx xX Va = /x> “x X h/ at = ab jogo +8" Jade = Ve J vve = We 1 AAT 2 P es Cs (eealeit —-q? RA tie: Sen the result being then multiplied by the The gth root of the pth power of x having been taken, the reciprocal of the result is raised to the 7th power, and its sth root is taken: this result having been first multiplied by the pth root of the qth power of a, is squared, and cube of x, the fourth root of the whole is taken. What is the algebraical me- thod of stating this process, and what power of x is the result ?, 1 meee |g Expression, 1 a ap ot 73 a L q apqs+2Qq78 —2p?r Result, 2@& sues ea oY xis ty aN ries ( «i 1B) =at2ahPh +b = 2 \F ae hel a P Rane AOS ee ee gt Pe OB 23 L3 a bs Can) Cass) =a—b us rk Me dlae Begins 2 a eciiiag: a a aCe ey aan n 90 z a : a a is a ea dink e) >a 3 ~1 4 ea ne a OS ah publ ess hs (Cpe ea 1 1 Estar ite Mette Neca, OP ria, A 22 gD? qetn-m Multiply together the two series EXAMPLES OF THE PROCESSES 4a al = 2 pi yi Oe =p) = a” b” ogre» br”. Ze a7 ™ pT eTY pee 1 1 fra = *) = a’ 3 2 1 4 ataae+ al g* + al" a + avatt ....5 and 64+ 0! a7! + Bi x? and give the terms of the product which involve ars ee ee Answers. (0.0 4 OO Or ie en ae ae Ge ry awe + OM gH 2 4 bw gmt a, , mM) SON Agee Cbha® + bf a®t” + 014) a (ab +a bv taliv t+ .....0 % (ad + a Bet 4 eo Ay L 1 2d a-b= (3 a ll +a) Ltt fa Beep anes a (at — ot) (43 att et | 1 1 3 ait a2 *) = (A - aD LG +a? b* + a* b? +b Lea) mn+ 1 pn+n—1l a BSD aor a Co HR he boy WE Steg ea a RO gS eM eo ve Nb b a” 6b a™ get? \/ aye) _ 4 ask vi DIS) IVA ae ‘ab ab SECTION 23. How much time elapses between two consecutive conjunctions of the minute hand and hour hand of a watch? Ans. 1 hour, 5 minutes, 27 seconds, and .8_ of a second. Tt If one hand revolved in a hours, and the other in & hours, what time would elapse between two conjunctions ? Ans. If 6 be greater than a, pom, hours. In the last question, how long will it be before the quicker hand has gained Miscellaneous Questions. Pp of a whole revolution upon the other? p ab Ans. hanes In ‘a+ ba | Be ets that if in the expression the expression itself be substituted fora, the result will be = 2, hours. a determine a and & so _Ans. a may have any value, pro- vided 6 = — 1, or the expression must a-x be : L te —————— OF ARITHMETIC AND ALGEBRA. 91 Show that in the series = howe, n—-1ln— 2 x nN T 2 3 the same result is produced by writing ceding series whenn=1 n=2 n=3 nm + 1 instead of 7, as would be pro- 2 = 4, and show that they are the same duced by multiplying the series by as 1+, (1+ 2)’, (1 + xv)*, and 1+. Write the values of the pre- (1 + 2)’. n ltneotn iin aR a ba Multiply together the two following series : x8 eS Ey ed ake th euch gre gigay &e. Pe ee eee a Dye oe mare cgi Omar: SN ee. and show that the product is the expression obtained by writing «+ y instead of a in the first. Multiply together the series atdatalet+ave tawatt+.... and O+ Bla + bl at + Olas + ON a +... What is the term of the product involving #”? Ans. (a b+ at ote a@ Dot el bof BOD 4 ag a" In the preceding question, let the second series be Tteotatt+ t+ att ween or let b=1 8=1 bi = 1 &es Find from the result an easy method of multiplying any series by the last men- tioned, and make use of it to find the first five terms of the fifth power of ltata’?+ 4x + &e. Ans. 1+ 5 2+ 152% + 35 a + 70 x4. Show that in the product of the two series atdatalaetal’ a + avx +... a—daet+a’e2—a" a + a at —....: there can be no terms involving odd powers of 2. a Show that the following are all equal to each other and to Tes a bea + ott Se. ri Sikes Ec oceeete | ie ot! 1-2 1 x 1 Three men, A, B and C, could complete a work as follows: A and Bin e days, B and Cin a days, A and ,C in d days. In what time could each complete if, and in what time could they all do it together? . Ans. A Band C could severally do it in ere ae Ga BE Oa: get ge AE Ce la ar abt+ac—be ab+bec—ac be +ac—ab ve —2abe and all three together in yeu eethea days. Explain the case in which a = 1, b = 4, andc = 6, and also that in which a=1, 6=2, c=2. Every whole number is either one of the series of powers of 2 contained in 1, 2, 4, 8, 16, &c., or may be made by adding together terms of this series without repeating any term twice. And every whole number is either one of the series 92 EXAMPLES OF THE PROCESSES of powers of 3 contained in 1, 3, 9, 27, 81, &c., or may be made by addition and subtraction of terms of this series without using any one twice. Prove the following formule: n(n + 1) 2 2 (2 + 1) (2% + 1) a 2 (n + 1° he 1 Tikes a +1)* +73 19° and having proved these, deduce from them the following theorems: 1. That the sum of all whole numbers up to 2 Res | is > n(m +1). 2. That the sum of _ m+) +2 2 _ @&@ +1) @ + 2) (2 + 3) xt Gee te woloy OuFe. b a WAN Ue the) 4 3 the squares of all whole numbers up to n* is : n(n + 1) (Qn + 1). 3. That the sum of the cubes of all whole num- bers up to 73, is the square of the sum of all whole numbers up to 7. From what immediately precedes, prove that the sum of 7 terms of the series a, at+b Go OM eg that the sum of 2 terms of nm — 1 is na +n — — 6 4 a 1 a (a+b? (a+ 2b).,.. isn@tnun—1ljab+ ; n(n—1) (2n —1) 02 and that the sum of 7 terms of a?, (a + 6)° &c., is 1 na? +5 n(n — Dab+ = n(n—1) (2n — Nab +s n* (nm — 1)? 6° "What is the inverse operation to ad- ding one mth part of the whole, and subtracting one th part of the whole? Answers. The subtraction of the (2 + 1)th part, and the addition of the (n — 1)th part. (The principal dif- ficulty is in the correct understanding of the words of the question.) There is a number to which I add its fourth part; from the sum I take 3, and to the difference I add its fifth part. The result is 10. What was the number? Ans, 9 Ha 15 There is a number to which a is added, and the result is divided by 8. To the quotient a’ is added, and the result divided by 8’. To the quotient a’ is added, and the result divided by b”, The result of the last process is found tobe. What was the number ? Ans. hb" b'b — al b—adb—a, In the preceding, let the addition be changed into a subtraction, and the division into multiplication. What is the number ? Ans. ; a! a’ . HCN da bi dk bi ae Multiply the expression a+ »/b + we by Wat V/b — ve, give the product the form P + ./Q, and mul- tiply by P — /Q. What is the result? Ans, @+ 82+C—2ab— 2bc— 2ca. How many different cases are there of + a +3, and what is the product of all ? Ans. (a2 - Be)" , How many different cases are there of tat b tc, and what is the pro- duct of all? — 2 Ans. (a + bt¢+¢4—2¢7°B— 28 - 2¢ a?) * Show that one of the three, a — 3, b—c, c — a, must be negative, and one must be positive ; and that a? + &?+ ¢? always exceeds ab + bc+ ca. If m be a given number, then x can always be taken so great that (a + m)?2 shall exceed 2* as much as we please ; and at the same time by as small a fraction of @ as we please, Show that the number of different ways (counting different orders as dif- ferent ways) in which p numbers, no one of which exceeds m, can be put together so as to make g, must be the co-efficient of x! in the development of OF ARITHMETIC AND ALGEBRA. 93 (e+ v2 + a+... tot aan)?, What are the roots of the following equation aww~etqetry + b(pe+qertnt+e=0? Ans. The four cases of —<$<$<<$<$< << —_ 3 ~ ga + aq? — 4apr—2bp+ 2p VP — 4a. 2 Va. p Show that the sum of these four roots is - a Prove the following formula by verification : V9 ph ee Fee = Mon +. Ve— be + Meo i ie For what numbers or fractions is w? — cy’ a square? Ans. m, n, and p, being any whole numbers or fractions, let espe mem) y = 2p nen. How must m, 2, and p, be taken, so that ¢ being a fraction, x* — cy’ may be a square whole number * - Assuming the following notation ie | 1 1 v= Dy 2Vizvre, 2V,= 2° +72) 2V, = & +s, &e. show that V,,, + V,.1 = 2V, Vi. Let p be a given whole number, and show that the following equation is satisfied by one value of z and m, and by one only ; mo"rti + 9% —~1l=2p +1. Form a series of terms beginning with 1, and such that each term exceeds the preceding by the cube of the units figure in the sum of all the preceding. Ans. 1, 2, 29, 37, 766, 891, &e. Show that the following equation, 2* = a x + bis verified by /' fy vee Je WE ies EIN oa Gee Fy WONG 2. ma Veta 8% and show that the product of the two terms in the value of « just given is : Find a value of P from the equations P = Q + a” P=1 + Qa, and show how this may be applied to deduce the following equation : aE eae Ve ot ett t+ cco He go from which deduce the following : y Se KD 2 yice ~ y" 2 aw ty? av + lee. f + a" oe y—-2&£ Detect the mistake in the following In the equations aw +by = ab process: . a+ y? = c*, what relation must exist Let a = b; then a? = a, orsa’ — between a, b, and ec, in order that the ab=0, and a = 0 or a — B= 0, resulting values of # may be equal? thence a2 — ab = a? — B’, or a (a—4) hed ab =(a+b) (a—b), or a=a-+-6. ite eg reg ane pee But 6 = a, thena =a+a= 2a. Va be If a and 6 be very nearly equal, then Two circles may cut each other in two points; two straight lines in one va — Nb point, and a straight line and circle in ——— = ly. ' a—6b 2/4 ma ee two points, How many different points 94 EXAMPLES OF THE PROCESSES’ of intersection may there be where is always positive when mand are there are 12 circles and 10 straight not less than 1. ‘ lines? There are three species of curves, Ans. 427. marked A, B, and C. Two of the sort A’may cut each other in a points, and two of the sorts B and C in 6 andec points. Again, A may cut B in ce! points, B may cut C ina’ points, and Anan Ses m-+-1 Cmaycut A in 8’ points. There are Pes y, ~~ i —)* m,n, and p curves of the three sorts: how many points of intersection may — Prove that the preceding expression there be in the figure ? Ans. What is the answer to the preceding, when there are m circles, and 7 straight lines ? 1 s(am® + bne +cep+2ainp+2U0pm+2dmn—am — bn —cp) Verify the following equations : 1 (vr +1) —@ 1.2 = (@+ 2)? — Q(a@ +1)? + a 1.2.3 = («+ 3)? — 3 («+ 2)3 + 3 (vw +1)8 — a3 1.2.3.4 = (4 + 4)* — 4 (@ + 3)4 + 6 (a+ 2)'—4(v4 1)4 + a4 And also the following, 0=(*#+2) -2(@v~+))+2 (0 = (x + 3)? —- 3 (a + 2)2+3 (v4 1)? - @& 0= (@+ 4)? — 4(@ + 3)? + 6(~@4+ 2)? — 4(@ + 122? + @? 0=(r+ 4)° — 4 (a+ 3)8+6 (w+ 2)? —4 (w@ + 1984+ 2 And also the following: ey = 2 + 29 ——— Il z—1 -— 2 2 3 ny oy | =a): £-lae-22-3 oe eee or OL Ee 2 3 4 2 3 4 If there be a series of terms a+ a’a2 + a! x + &ce., of which the co- efficients a, a’, a” &c. follow this law, namely, that each one, after the second, is the sum of the two preceding, then if V represent the sum of the series ad infinitum, we must have a+ (a —a)x l—a—a’ : and if V, represent the sum of the series as far as a” 2” inclusive, we must have Vn be Bel | aR IES eB sak BY at x—1 ANT Rite ane +_36 @ Ves at (a! —a)av = a@ty) gt_ q® grt? fi l1—-g— 2 Show thata—6+c—e-+ .... must be less than a, and greater than a — 6, if a, 6, c, &e., be a series of decreasing positive quantities. Reduce the binomial theorem n n— i —~|l n= Qt ay sltnahnewe B+ nS enh wy i, to the following form: 1 a a? a3 (i+anry) = era Sie ate @! mA) or eee (liom G) fl —9 gi +70... OF ARITHMETIC AND ALGEBRA, 95 What expressions are those, which, substituted instead of x in the following, 1-2 x — x 1. —— 2%anr+ 2 Arn apy 4,a(e@ +m) =—m™ will reduce them severally to a. Answers, 1—2z a 2 ve Ne x 1, —— Fe Neh eid see 627 & 4. = (@ +m) =m, 12 What expression must be substituted for # in toe eee in order that it may become 6+ x? Ans. 9 If the seriesa t+a’x + ala? + .... be reduced to the form a (1 + px + pqa+pqra + ....), what are p, 4g, 7, &e.? If the expression a? + wy + 7? change from 2 to 10 when w changes from 1 to 2, what are the changes of y? What. is the least number or fraction by which 7 more than a square number or fraction can exceed 5 times the number or fraction itself? Ans. = 4 2 Find the sum of the squares of the roots of v* — (1 ta) a+ rere, without finding the roots. Ans. a. Verify the last by finding and squaring the roots. What is the expression which has for its roots the squares of the roots of az*-+ ba +c? Ans. @x2+ Qac—P)x+ ce. Divide the number a into two such parts, that the first shall be the square of the second, Mitek a Af 17 hae ag oe tI 1+4a ied is 1 Show that if a be a whole square number, the answer of the last must be irrational, and also that the answer cannot be rational unless @ be of the form 6 (6+ 1) where 0 is rational. Divide the number a into two parts, the product of which shall be a square. m? a n? a yee meas Bg ey aa a where m and 7 are any whole numbers. Show that these parts cannot be whole numbers unless a or one of its factors be the sum of two square numbers. Divide the number a into two such parts that the sum of their squares shall be a square. (n? — m*)a ‘ 2mna n?+2mn — m oY n?+2mn —m where m and m are any numbers, 2 being the greater. Show that these parts may be made whole numbers when a is a whole number, itself or one of its fac- tors béing the difference between one square number and the double of another. Solve the equations Ans, ] ] 1 sf + B = a4 2 +-—= 5 senp i=) 65 £ y y z z x and explain the solution when a + 6 = D6 PROCESSES OF ARITHMETIC AND ALGEBRA. _ << ). There are 2 numbers, the sum of all but the first is q, of all but the second, : a,, &e. &e., and of all but the last, a, What are the numbers? Ans. The first is — 1 : ary (ib Gy ithidat 3. a ane Os - the second is ——— (ai + a + oy + eoee TF An) — A and so on. If there be three whole numbers, the product of every two of which is a square, then the numbers themselves must be squares. Required the least number, which, divided by 4, 6, or 9, leaves a remainder 3, and by 15, a remainder 12. Avs. 147. . Of the four numbers, x wy xy* wy’, in continued proportion, the sum of the first and last is 6, of the second and third a. Required # and y. a+ ®ANP 4 20) oe Ans. y = on 4 a SC ts oe ee Se eee (atb+tvP 4206 - 3a) Batbt VP+ 206-30 The upper or lower sign being used throughout. Explain the two solutions, show that of the two values of y, each is unity divided by the other. Show that the preceding results are rational when, m and m being any whole numbers, 37n2+m a n(n—m) 2 Ina, vy, ry, vy, let the sum of the first and third be p, and that of the second and fourth gq. Required # and y. b= 3 Ans. y= 4 aie _P se p Pate: a ‘Cc é £ Let c= ——— = ——— = r= =, btp P a+q FT FF h Required the value of x, so that p, g, and 7 shall not appear in it. a @ (dfh+dgtehja c _bdfh+bdg + beh’ + cfh + ce OT al ae g osetia Explain this in the case where dfh + dg -—+ eh = 0. Show that the following equations may all be satisfied by a value of # which is less than unity. (ath)? = @&+2(a+2).h (ath? ='8 + 3(a + x)*.h (atéAp=A8+3e@VA+ 38(at+ah’ (a+th)t = at + 3(A4+2)°.h q fir Te ns 1 h Va + ‘ne Vo ATE apron a Ans, Printed by Wint1am CLowezs and Sons, Stamford-street, Gs ee ‘ 4 'd i vi) Ae. ub | im phi eral Ry : ee: Ae a iy ao ay 4k Ti nem a / ay a | Mra! CA UNIVERSITY OF ILLINOIS-URBANA 510L61 C001 vo01 | LIBRARY OF USEFUL KNOWLEDGE. MATHEMATIC i Hl | | I |