ft^- (^ (3 3} ff^f^l y wm ^^A^ 1 / ^^^^ / "^^ g ^a-i^ftaoTT^ "\ 6^W"/P9j^ Digitized by tine Internet Arciiive in 2010 witii funding from Boston Library Consortium IVIember Libraries littp://www.arcliive.org/details/natureutilityofmOOdavi -^/^z-/ U>-t.<,-'-^^i^ ^o-n.^ <:/ ^;^^ tL^^<^ THE Nature and Utility OF MATHEMATICS, WITH THE BEST METHODS OF INSTRUCTION EXPLAINED AND ILLUSTRATED BY CIIAKLES DA VIES, LL.D., EMERITUS PKOFESSOR OP HIGHER MATHEMATICS IN COLUMBIA COLLEOB. loavoH cQii^^^ ]:-^^' CHESTNUT HUX, MA. NEW YORK: PUBLISHED BY A. S. BARNES & CO., Ill AND 113 William Street. 1873. BAVIES' MATHEMATICS. IN THRKE PARTS. I-GOMMOIT SCHOOL COURSE. ©avies' Primary Aritliiiietic— The fundamental principles clisplaycd in Object Lessons. Eavios' Iiitellectsial Aritliiiietic— Eereniii, science. easily remembered, readily referred to, and ad- vantageously applied, will alone suffice to sift the pure metal from the dust of ages, and fashion it for present use. Such knowledge is Science. PLAN OF THE WORK. 21 Masses of facts, like masses of matter, are ca- Knowledge pabie 01 very mmute subdivisions; and when we aucedtoita know the law of combination, they are readily ^^'^"»'^°^ divided or reunited. To know the law, in any case, is to ascend to the source ; and without that knowledge the mind gropes in darkness. It has been my aim to present such a view objects of of Logic and Mathematical Science as would clearly indicate, to the professional student, and even to the general reader, the outlines of these subjects. Logic exhibits the general formula i^ogicaiwi 1-11 11 1 • 1 r • 1 mathematica applicable to ail kinds oi argumentation, and mathematics is an application of logic to the abstract quantities Number and Space. When the professional student shall have ex- amined the subject, even to the extent to which certainty of it is here treated, he will be impressed with the clearness, simplicity, certainty, and generality of its principles ; and will find no difficulty in ma- king them available in classifying the facts, and examining the organic laws which characterize his particular department of knowledge. Thirdly. Mathematical knowledge differs from Maihemati- every other kind of knowledge in this : it is, as ^^ ^Z^ it w'ere, a web of connected principles spun out from a few abstract ideas, until it has become one of the great means of intellectual develop- its extent. 22 INTRODUCTION. ment and of practical utility. And if I am per- Necessity mitted to cxtcnd the figure, I may add, that the of beginning t r \ • i i i at the right WBD of the spidcr, though perfectly simple, if we *''"*■ see the end and understand the way in which it is put together, is yet too complicated to be unravelled, unless we begin at the right point, and observe the law of its formation. So with mathematical science. It is evolved from a few — a very few — elementary and intuitive princi- How pies : the law of its evolution is simple but ex- mathemati- . i i • i • i i cai science ifl acting, and to bcgm at the right place and pro- constructe . ^gg^ jj^ ^jjg j-jght way, is all that is necessary to make the subject easy, interesting, and useful. What has I have endeavored to point out the place of been at- _ _ templed, beginning, and to indicate the way to the math- ematical student. I am aware that he is start- ing on a road where the guide-boards resemble each other, and where, for the want of careful observation, they are often mistaken ; I have sought, therefore, to furnish him with the maps and guide-books of an old traveller. Advantages By explaining with minuteness the subjects of examining , i • i i • i • the whole about which mathematical science is conversant, subject, ^j^g whole field to be gone over is at once sur- veyed : by calling attention to the faculties of Advantages \\^q mind which the science brings into exercise, of consider- "^ ing the men- we are better prepared to note the intellectual tal faculties : ^ . i • t t • 11 operations which the processes require ; and by PLAN OF THE WORK. 23 a knowledge of the laws of reasoning, and an ofaknowj- . , , CI ^t^se of the acquaintance with the tests oi truth, we are en- lawsofrea- abled to verify all our results. These means have s°'»'^- been furnished in the following work, and to aid the student in classification and arrangement, diagrams have been prepared exhibiting separ- What has been doatii ately and in connection all the principal parts of mathematical science. The student, therefore, who adopts the system here indicated, will find his way clearly marked out, and will recomiise, Advantage* '' J o ' tothesto- from their general resemblance to the descrip- dent, lions, all the guide-posts which he meets. He will be at no loss to discover the connection between the parts of his subject. Beginning with first principles and elementary combina- tions, and guided by simple laws, he will go for- ^^'•e^ he begins. ward from the exercises of Mental Arithmetic to the higher analysis of Mathematical Science on an ascent so gentle, and with a progress so omer of progress steady, as scarcely to note the changes. And indeed, why should he ? For all mathematical processes are alike in their nature, governed by the same laws, exercising the same faculties, unity of and lifting the mind towards the same eminence. the Bubjeot, Fourthly. The leading idea, in the construe- Advantages to the tion of the work, has been, to afford substantial professional aid to the professional teacher. The nature ol '<'^*>«'- 24 INTRODUCTION, His duties: his duties — their inherent difficulties — the per- Discouragc- plexities which meet him at every step — the want difflcuiues: t>f Sympathy and support in his hours of discour- agement — (and they are many) — are circum- stances which awaken a lively interest in the hearts of all who have shared the toils, and been themselves laborers in the same vineyard. He takes his place in the schoolhouse by the road- side, and there, removed from the highways of Bemotenesa life, Spends his days in raising the feeble mind life. of childhood to strength — in planting aright the seeds of knowledge — in purbing the turbulence of passion — in eradicating evil and inspiring good. The fruits of his labors are seen but once in a generation. The boy must grow to fruits of manhood and the girl become a matron before hia efforts, . • i i • i i i i when seen he IS ccrtam that his labors have not been m vain. Yet, to the teacher is committed the high trust of forming the intellectual, and, to a certain ex- tent, the moral development of a people. He Theimpor- holds in liis hands the keys of knowledge. If labors. ^^® ^^'^^ moral impressions do not spring into life at his bidding, he is at the source of the stream, and gives direction to the current. Al- though himself imprisoned in the schoolhouse, his influence and his teachings affect all condi- tions of society, and reach over the whole hori- PLAN OF THE WORK. 25 zon of civilization. He impresses himself on The influence the young of the age in which he lives, and lives again in the age which succeeds him. All good teaching must flow from copious scurcesof good teach- knowledge. The shallow fountain cannot emit ing. a vigorous stream. In the hope of doing some- thing that may be useful to the professional teacher, I have attempted a careful and full objects for which the analysis of mathematical science. I have spread work was undcrtakeo. out, in detail, those methods which have been carefully examined and subjected to the test of long experience. If they are the right meth- principles ods, they will serve as standards of teaching ; .° \ ^^^ 'J *=> ' ing, the same for, the principles of imparting instruction are the same for all branches of knowledge. The system which I have indicated is com- System, plete in itself It lays open to the teacher the entire skeleton of the science — exhibits all its ^'^hat u presents. parts separately and in their connection. It explains a course of reasoning simple in itself, what u and applicable not only to every process in **p'''"* mathematical science, but to all processes of argumentation in every subject of knowledge. The teacher who thus combines science with science art, no longer regards Arithmetic as a mere <»'nWned ^ ° with art: treadmill of mechanical labor, but as a means — 26 INTRODUCTION. Theadvan- and the simplest means — of teaching the art and taefts result- . r • a.- t. i ii_* • iM from it. science of reasonmg on quantity — and this is the logic of mathematics. If he would accom- Resuiuof piish well his work, he must so instruct his uon. pupils that they shall apprehend clearly, think quickly and correctly, reason justly, and above all, he must inspire them with a love of knowl- edge. BOOK I. LOGIC. CHAPTER I. OEFINiTION'S— OPERATIONS OF THE MIND ^TERMS DEFINKD. DEFINITIONS. § 1. Defivition is a metaphorical word, which Definition a literally signifies " laying down a boundary." metaphorical All definitions are of names, and of names only ; Some but in some definitions, it is clearly apparent, definitions that nothing is intended except to explain the ''oniy'° meaning of the word; while in others, besides '^o'^'^- explaining the meaning of the word, it is also ^;j,„^j implied that there exists, or may exist, a thinff «>'"respond- ^ ' "^ =• ingtothe corresponding to the word. words. § 2. Definitions which do not imply the exist- or definitions ence of things corresponding to the words de- not imply fined, are those usually found in the Dictionary """^ ^°"^ •' J spondmg of one's own language. They explain only the to words. 28 LOGIC. [book I. ,j. meaning of the word or term, by giving some explain equivalent expression which may happen to be words by ' ^ ^ ri equivalents, better known. Definitions which imply the ex- istence of things corresponding to the words de- fined, do more than this. For example : " A triangle is a rectilineal fig- ure having three sides." This definition does two things : 1st. It explains the meaning of the word tri- angle ; and, 2d. It implies that there exists, or may exist, a rectilineal figure having three sides. Definition of a triangle ; what it implies. ofa § 3. To define a word when the definition is definition which im- to imply the existence of a thing, is to select ^ iTence of' fi'O"^ ^11 the properties of the thing those which a thing. „ j,g i^Qgt simple, general, and obvious ; and the Properties properties must be very well known to us before known. ^^® ^^^ decide which are the fittest for this pur- pose. Hence, a thing may have many properties besides those which are named in the definition A definition of the word which stands' for it. This second truth. l^ii^^ o^ definition is not only the best form of ex- pressing certain conceptions, but also contributes to the development and support of new truths. to § 4. In Mathematics, and indeed, in all strict Mathematics names imply scienccs, namcs imply the existence of the things CHAP. I.] DKFINITIONS. 29 which they name ; and the definitions of those things names express attributes of the things ; so that express ,'£••.• I , c iU attributes no correct aennition whatever, oi any mathe- matical term, can be devised, which shall not express certain attributes of the thing correspond- ing to the name. Every definition of this class definitions is a tacit assumption of some proposition which °''""^<^i'^' is expressed by means of the definition, and propositions. which gives to such definition its importance. § 5. All the reasonings in mathematics, which Keasoning rest ultimately on definitions, do, in fact, rest /'f"."^°" •' denmtiona; on the intuitive inference, that things corre- rests on spondmg to the words defined have a conceiv- intuitive able existence as subjects of thought, and do or may have proximately, an actual existence.* * There are four rules which aid us in framing defini- p-our rules tions. 1st. The definition must be adequate: that is, neither too 1st rule, extended, nor too narrow for the word defined. 2d. The definition must be in itself plainer than the word id rule, defined, else it would not explain it. 3d. The definition should be expressed in a convenient -^^ ^iig, number of appropriate woi-ds, 4th. When the definition implies the existence of a thing corresponding to the word defined, the certainty of lliat existence must be intuitive. 30 LOGIC. [book I. OPERATIONS OF THE MIND CONCERNED IN REASONING. Three opera- § Q There are three operations of the mind dons of the which are immediately concerned in reasoning. 1st. Simple apprehension ; 2d. Judgment ; 3d. Reasoning or Discourse. Sim lea ^ '^- Simple apprehension is the notion (or prehension, conception) of an object in the mind, analogous to the perception of the senses. It is either incompicx. Incomplcx or Complex. Incomplex Apprehen- sion is of one object, or of several without any relation being perceived between them, as of a Complex, triangle, a square, or a circle : Coiuplex is ot several with such a relation, as of Zi triangle within a circle, or a circle within a siju^rc. § 8. Judgment is the compariur; together in the mind two of the notions Cor ideas) which Judgment ^ defined, are the objects of apprehension, whether com- plex or incomplex, and pronouncing that they agree or disagree with each other, or that one of them belongs or does not belCj/ to, the other: for example : that a right-arg'od t-^iangle and an Judgment equilateral triangle belong to thf> r'^ass of figures either Called triangles ; or that a iou'.rP is not a circle, rma i\e j^(jgfyj,gjj^^ therefore, is eitiipr Affirmative or Neg negative. ^^j-j,g CHAP. 1 J ABSTRACnON, 31 § 9. Reasoning (or discourse) is the act of neasoning proceeding from certain judgments to another founded upon them (or the result of them). §10. Language affords the signs by which Language aSb rds signs of these operations of the mind are recorded, ex- pressed, and communicated. It is also an in- thought: strument of thought, and one of the principal also, an instrument helps in all mental operations ; and any imper- of thought, fection in the instrument, or in the mode of using it, will materially affect any result attained through its aid. § 11. Every branch of knowledge has, to a Every branch certain extent, its own appropriate language ; ofknowiedge 1 r -1 -1 1 • I has its own and lor a mmd not previously versed m the language, meaning and right use of the various words and signs which constitute the language, to attempt must be learned. the study of metiiods of philosophizing, would which be as absurd as to attempt reading before learn- ing the alphabet. ABSTRACTION. § 12. The faculty of abstraction is that power of the mind which enables us, in contemplating any object (or objects), to attend exclusively to Abstraction, 32 LOGIC. [book I, some particular circumstance belonging to it, and quite withhold our attention from the rest. Thus, in . contempia- if a pcrson in contemplating a rose should make " " ' the scent a distinct object of attention, and lay aside all thought of the form, color, &c., he would draw off", or abstract that particular part ; of drawing and therefore employ the faculty of abstraction. He would also employ the same faculty in con- sidering whiteness, softness, virtue, existence, as entirely separate from particular objects. § 13. The term abstraction, is also used to denote the operation of abstracting from one or The term -' ° tbstraciion, more things the particular part under consider- hc'v used. . . ation ; and likewise to designate the state oi the mind when occupied by abstract ideas. Hence, abstraction is used in three senses : Abstraction ^^^- '^^ denote a faculty or power of the ^?"T mind ; a faculty, ' a process, gd. To dcnotc a process of the mind ; and, and a state * of mind. 3(j^ To denote a state of the mind. GENERALIZATION. General Iza- § 14. Generalization is the process of con- tion— the templating the agreement of several objects in process of contempia- certain points (that is, abstracting the circum- ting the _ _ -j«r.'(!ment. stauccs of agreement, disregarding the diner- CHAP. I.] TERMS. 33 of several things. ences), and giving to all and each of these ob- jects a name applicable to them in respect to this agreement. For example ; we give the name of triangle, to every rectilineal figure hav- ing three sides : thus we abstract this property P , ... Generaliza- from all the others (for, the triangle has three uon angles, may be equilateral, or scalene, or right- angled), and name the entire class from the prop- erty so abstracted. Generalization therefore imphM ., . ,. ■. . , , , abstraction. necessarily implies abstraction ; tnough abstrac- tion does not imply generalization. TERMS SINGULAR TERMS COMMON TERMS. § 15. An act of apprehension, expressed in language, is called a Term. Proper names, or a term, any other terms which denote each but a single individual, as " Caesar," " the Hudson," " the Conqueror of Pompey," are called Singular singular terms. Terms. On the other hand, those terms which denote any individual of a whole class (which are form- ed by the process of abstraction and generaliza- tion), are called Common or general Terms. For commoD termd. example ; quadrilateral is a common term, appli- cable to every rectilineal plane figure having four sides ; River, to all rivers ; and Conqueror, to all conquerors. The individuals for which a common term stands, are called its Signijicates. signiflcates 3 34 LOGIC. [book 1. CLASSIFICATION. Genus, species. Examples classification. ^ -a .• § 16. Common terms afford the means of clas- Classiflcation. ' sification ; that is, of the arrangement of objects into classes, with reference to some common and distinguishing characteristic. A collection, com- prehending a number of objects, so arranged, is called a Genus or Species — genus being the more extensive term, and often embracing many species. For example : animal is a genus embracing every thing vv^hich is endowed with life, the pow- er of voluntary motion, and sensation. It has many species, such as man, beast, bird, &c. II we say of an animal, that it is rational, it be longs to the species man, for this is the charac- teristic of that species. If we say that it has wings, it belongs to the species bird, for this, in like manner, is the characteristic of the species bird. A species may likewise be divided into classes, or subspecies ; thus the species man, may be divided into the classes, male and female, and these classes may be again divided until we reach the individuals. Subspecies Principles § 17. Now, it will appear from the principles of jiassification. which govcm this system of classification, that '. 1.] CLASSIFICATION. 35 the characteristic of a genus is of a more exten- cenusmore .^ . , . , ^ . extensive sive signihcation, but involves lewer particu- th;m spodea, lars than that ot a species. In Hke manner, the characteristic of a species is more extensive, but less full and complete, than that of a subspecies ^"' i^ss fuii or class, and the characteristics of these less full complete. than that of an individual. For example ; if we take as a genus the Quadri- laterals of Geometry, of which the characteristic is, that they have four sides, then every plane rectilineal figure, having four sides, will fail under this class. If, then, we divide all quadrilaterals into two species, viz. those whose opposite sides, taken two and two, are not parallel, and those whose opposite sides, taken two and two, are parallel, we shall have in the first class, all irreg- ular quadrilaterals, including the trapezoid (1 and 2) ; and in the other, the parallelogram, the rhom- bus, the rectangle, and the square (3,4, 5, and 6). If, then, we divide the first species into two subspecies or classes, we shall have in the one, the irregular quadrilaterals (1), and in the other, the trapezoids (2) ; and each of these classes, being made up of individuals having the same char- acteristics, are not susceptible of further division. If we divide the second species into two classes, arranging those which have oblique an- gles in the one, and those which have right 36 LOGIC. [book 1. Species and classes. angles in the other, we shall have in the first, two varieties, viz. the common parallelogram and the equilateral parallelogram or rhombus (3 and 4) ; and in the second, two varieties also, viz. the rectangle and the square (5 and 6). Now, each of these six figures is a quadri- lateral ; and hence, possesses the characteristic of the genus ; and each variety of both species Each indi- vidual falling under tlie genus enjoys all the enjoys all the characteristics of the species to characteris- tics. which it belongs, together with some other dis- tinguishing feature ; and similarly, of all classi- fications. Subaltern genus. Parallelo- gram. § 18. In special classifications, it is often not necessary to begin with the most general char- acteristics; and then the genus with which we begin, is in fact but a species of a more extended classification, and is called a Subaltern Genus. For example ; if we begin with the genus Par- allelogram, we shall at once have two species, viz. those parallelograms whose angles are oblique and those whose angles are right angles ; and in each species there will be two varieties, viz. in the first, the common parallelogram and the rhom- bus ; and in the second, the rectangle and square. § 19. A genus which cannot be considered Highest ° genus. as a species, that is, which cannot be referred CHAP. I.] NATURE OF COMMON TERMS. 37 to a more extended classification, is called the Highest highest genus ; and a species which cannot be Lowest considered as a genus, because it contains only species, individuals having the same characteristic, is called the lowest species. NATURE OF COMJIGN TERMS. § 20. It should be steadily kept in mind, that the " common terms" employed in classification, have not, as the names of individuals have, any A common term has no real thing real existing thing in nature corresponding to co''i''«po'Jd- them ; but that each is merely a name denoting a certain inadequate notion which our minds inadequate have formed of an individual. But as this name does not include any thing wherein that indi- does not . ^-, . include any vidual diners from others of the same class, it thing in is applicable equally well to all or any of them, individuals Thus, quadrilateral denotes no real thing, dis- '^'^'"'» tinct from each individual, but merely any recti- lineal figure of four sides, viewed inadequately ; that is, after abstracting and omitting all that is peculiar to each individual of the class. By ^ ■^ but is this means, a common term becomes applicable applicable to alike to any one of several individuals, or, taken individuals. in the plural, to several individuals together. Much needless difficulty has been raised re- „ ^, -' Needless spec ting the results of this process : many hav- difficulty. ing contended, and perhaps more having taken 38 LOGIC. [book I, Difficulty in it for granted, that there must be some really the interpre- . . , . ,. i /■ ^i tationof existmg thing corresponding to each ol those 'T™-™"" common terms, and of which such term is the name, standing for and representing it. For ex- ample ; since there is a really existing thing cor- Noone responding to and signified by the proper and real thing gj^jg^jg^j. ^amc " iEtua," it has been supposed correspond- o ^ ^ ins to each. |]-^^|. the common term "Mountain" must have some one really existing thing corresponding to it, and of course distinct from each individual mountain, yet existing in each, since the term, being common, is applicable, separately, to every one of them. The fact is, the notion expressed by a common term is merely an inadequate (or incomplete) Merely an j^q^jqj^ q|- ^^j individual ; and from the very cir- inadequate •' notion pai- cumstaucc of its inadequacy, it will apply equally tiallyde- i ■j . i ^ x ^ signaling ^ygU ^q any ouc of scvcral individuals. For ex- the thing. ample ; if I omit the mention and the consider- ation of every circumstance which distinguishes iEtna from any other mountain, I then form a notion, that inadequately designates iEtna. This "Mountain" Hotion is expressed by the common term " moun- ** tain," which does not imply any of the peculiar- applicable ' I. J J I ^oaii ities of the mountain ^Etna, and is equally ap- tnountains. plicable to any one of several individuals. In regard to classification, we should also bear in mind, that we may fix, arbitrarily, on the CHAP, ij SCIENCE. 39 characteristic which we choose to abstract and May axon attributes consider as the basis of our classification, disre- arbitraruy garding all the rest : so that the same individual giassjflcation may be referred to any of several different spe- cies, and the same species to several genera, as suits our purpose. SCIENCE. § 21. Science, in its popular signification, means knowledge.* In a more restricted sense, science in its general it means knowledge reduced to order ; that is, sense. knowledge so classified and arranged as to be easily remembered, readily referred to, and ad- Has a vantageously applied. In a more strict and gigi^fl"ation. technical sense, it has another signification. " Every thing in nature, as well in the in- animate as in the animated world, happens or is done according to rules, though we do not always know them. Water falls according to the laws of gravitation, and the motion of walk- ®°'^'^ "^^^ ing is performed by animals according to rules. The fish in the water, the bird in the air, move according to rules. There is nowhere any want of rule. When we think we find that want, we Nowhere can only say that, in this case, the rules are un- any warn of rule. known to us. f Assuming that all the phenomena of nature Views of K>mt. * Section 23. f Kant. 40 LOGIC. [bock I Science are consequences of general and immutable laws, a technical we may define Science to be the analysis of sense e ne . ^j^^g^ laws, — Comprehending not only the con- an analysis nected Drocesses of -experiment and reasoning of the laws r r o of nature, which make them known to man, but also those processes of reasoning which make known their individual and concurrent operation in the de- velopment of individual phenomena. § 22. Art is the application of knowledge to .vrt, practice. Science is conversant about knowl- upphcation q^^^q ■ ^j.^ jg ^he usc or application of knowl- of ^ ^ ^ science, edge, and is conversant about works. Science has knowledge for its object : Art has knowledge for its guide. A principle of science, when ap- phed, becomes a rule of art. The developments of science increase knowledge : the applications of art add to works. Art, necessarily, presup- presupposes pQses knowledge : art, in any but its infant state, knowledge. o ^ presupposes scientific knowledge ; and if every art does not bear the name of the science on which it rests, it is only because several sciences are often necessary to form the groundwork oi a single art. Such is the complication of hu- mustbe^* man afiairs, that to enable one thing to be done, known be- ore one can be done, erties of many things. it is often requisite to know the nature and prop fore one can CHAP. II.] KNOWLEDGE. 41 CHAPTER II. SOURCES AND MEANS OF KNOWLEDGE — INDUCTION KNOWLEGDE. § 23. Knowledge is a clear and certain con- Knowledge j^ , .... .... , a clear con- ception 01 that which is true, and implies three ceptionof things : '^*"'' ^' "'"'' 1st. Firm belief; 2d. Of what is true; and, impiies- 3d. On sufficient grounds. belief- If any one, for example, is in doiiht respecting -''• ^fwhat one of Legendre's Demonstrations, he cannot 3d. on be said to know the proposition proved by it. If, ^"^ *^"^" r r r j grounds. again, he is fully co7ivinced of any thing that is not true, he is mistaken in supposing himself to know it ; and lastly, if two persons are each fullj/ conjidenl, one that the moon is inhabited, and the other that it is not (though one of these opinions must be true), neither of them could properly be said to know the truth, since he cannot have sufficient proof of it. Examples. 4'.4 LOGIC. [book I FACTS AND TRUTHS. „ ... § 24. Our knowledare is of two kinds : of facts Knowledge is ^ O of facts and j^^j-^j truths. A fact is anv thina; that has been truths. •' ° or IS. That the sun rose yesterday, is a fact : that he gives Kght to-day, is a fact. That wa- ter is fluid and stone soUd, are facts. We de- rive our knowledge of facts through the medium of the senses. Truth an Truth is an exact accordance with what has accordance with what BEEN, IS, Or SHALL BE. There are two methods has been, is, ^ , . . , , or shall be. o^ asccrtaming truth : Two methods of ascertain- Ist. By comparing known facts with each ins it. other; and, 2dly. By comparing known truths with each other. Hence, truths are inferences either from facts or other truths, made by a mental process called Reasoning. § 25. Seeing, then, that facts and truths are the Facts and elements of all our knowlcd2;e, and that knowl- truths, the ^ elements edge itsclf is but their clear apprehension, their knowledge, fii'm belief, and a distinct conception of their relations to each other, our main inquiry is. How are we to attain unto these facts and truths, which are the foundations of knowledge ? 1st. Our knowledge of facts IS derived through CHAP. II.] FACTS AND TRUTHS. 43 the medium of our senses, by observation, exper- iment,* and experience. We see the tree, and How we s.rrive at a perceive that it is shaken by the wind, and note knowledge oi the fact that it is in motion. We decompose w^ater and find its elements ; and hence, learn from experiment the fact, that it is not a simple substance. We experience the vicissitudes of heat and cold ; and thus learn from experience that the temperature is not uniform. The ascertainment of facts, in any of the vi^ays above indicated, does not point out any connec- This does not tion between them. It merely exhibits them to connection the mind as separate or isolated : that is, each ^^^^^'^^ ^ ' them, as standing for a determinate thing, whether simple or compound. The term facts, in the sense in which we shall use it, will designate facts of this class only. If the facts so ascer- tained have such connections with each other, when they that additional facts can be inferred from them, nectionthat that inference is pointed out by the reasoning 'b''°h|fre°"' process, which is carried on, in all cases, by com- ®°"'"" process. parison. 2Jly. A result obtained by comparing facts, we Truth, ro-md have designated by the term Truth. Truths, ^7001^^^""^ therefore, are inferences from facts ; and every * Under this term we include all the methods of inves- tigation and processes of arriving at fac ts, except the pro- cess of reasoning. 44 LOGIC. [book 1 and is inferred from tliern. How truth has reference to ail the singular facts from which it is inferred. Truths, therefore, are re- sults deduced from facts, or from classes of facts. Such results, when obtained, appertain to all facts of the same class. Facts make a genus : truths, a species ; with the characteristic, that they be- come known to us by inference or reasoning. § 26. truths are j- /• x i_ ^l • n inferred from ifom lacts by the rcasoumg process r reasonin; process, Ist case. How, then, are truths to be inferred There facUby the ^^g ^^^ ^^ggg_ 1st. When the instances are so few and simple that the mind can contemplate all the facts on which the induction rests, and to which it refers, and can make the induction without the aid of other facts ; and, 2dly. When the facts, being numerous, com- plicated, and remote, are brought to mind only by processes of investigation. Qd case. INTUITIVE TRUTH. Intuitive Self-evident truths. Intuition defined. § 27. Truths which become known by con- sidering all the facts on which they depend, and which are inferred the moment the facts are apprehended, are the subjects of Intuition, and are called Intuitive or Self-evident Truths. The term Intuition is strictly applicable only to that mode of contemplation in which we look at CHAP. II.] INTUITIVE TRUTH. 45 facts, or classes of facts, an J apprehend the relations of those facts at the same time, and by the same act by which we apprehend tlie facts themselves. Hence, intuitive or self-evi- How intuitive 1 I I 1-1 • I • Irulhsare dent truths are those which are conceived in conceived in the mind immediately ; that is, which are per- ^'^ ™" " fectly conceived by a single process of induc- tion, the moment the facts on which they depend are apprehended, without the inteivention of other ideas. They are necessary consequences of conceptions respecting which they are asserted. Axioms of The axioms of Geometry afford the simplest and i^l^iJL^l^ most unmistakable class of such truths. '''"'^' "A whole is equal to the sum of all its parts," a whole , f. . , t • r T I- equal to the IS an intuitive or seli-evident truth, interred irom g„mofaii facts previously learned. For example ; having ^^'^. ^^.^ I •/ -i ' o an intuitive learned from experience and through the senses '^'^*- what a whole is, and, from experiment, the fact that it may be divided into parts, the mind per- ceives the relation between the whole and the sum of the parts, viz. that they are equal ; and then, by the reasoning process, infers that the now inferred, same will be true of every other thing; and hence, pronounces the general truth, that "a whole is equal to the sum of all its parts." Here all the facts from which the induction is drawn, -*" ^e facts are presented are presented to the mind, and the induction to the mind, is made without the aid of other facts ; hence, 4G LOGIC. , [book 1. Au the it is an intuitive or self-evident truth. All the deduced in Other axioms of Geometry are deduced Irom the same . , , ^ . ^ . , ^ay_ premises and by processes oi mierence, entn'ely similar. We would not call these experimental truths, for they are not alone the results of ex- periment or experience. Experience and exper- iment furnish the requisite information, but the reasoning power evolves the general truth. " When we say, the equals of equals are equal, we mentally make comparisons in equal spaces, These equal times, &c. ; so that these axioms, how- axioms are ,^ . , ■^^ i general cvcr sclf-cvidcnt, are still general propositions : propositions. ^^ ^^^ ^^ ^^^ inductivc kind, that, independently of experience, they would not present themselves to the mind. The only difference between these and axioms obtained from extensive induction is Diflference this : that, in raising the axioms of Geometry, them^and ^^^ instauccs offcr thcmsclves spontaneously, and other without the trouble of search, and are few and propositions, which re- simple : in raisins; those of nature, they are in- quire diugent 1 o ^ research, finitely numerous, complicated, and remote ; so that the most diligent research and the utmost acuteness are required to unravel their web, and place their meaning in evidence."* * Sir John Herschel's Discourse on the study of Natural Philosophy. CHAP. II.] LOGICAL TRUTHS. 47 TRUTHS, OR LOGICAL TRUTHS. ^ 28. Truths inferred from facts, by the process of generalization, when the instances do not offer Truths 11 1 1 ■ 1 1 X ■ generalized themselves spontaneously to the mind, but require from facta, search and acuteness to discover and point out t^iu^sin- their connections, and all truths inferred from ferredtrom truths. truths, might be called Logical Truths. But as we have given the name of intuitive or self- evident truths to all inferences in which all the facts were contemplated, we shall designate all others by the simple term, Truths. It might appear of httle consequence to dis- Necessity or tinguish the processes of reasoning by which ii^edistinc- truths are inferred from facts, from those in which the basis of :i classification. we deduce truths from other truths ; but this dif- ference in the premises, though seemingly slight, is nevertheless very important, and divides the subject of logic, as we shall presently see, into two distinct and very different branches. § 29. Logic takes note of and decides upon Logic the sufficiency of the evidence by which truths sufficiency of are established. Our assent to the conclusion evidence. being grounded on the truth of the premises, we never could arrive at any knowledge by rea- soning, unless something were known antece- dently to all reasoning. It is the province of Hs provinctt, 48 LOGIC. [book I. Furnishes Logic to fumish the tests by which all truths the testa of , ... i • r i r i truth. that are not intuitive may be inierred irom the premises. It has nothing to do with ascertain- ing facts, nor with any proposition which claims to be believed on its own intrinsic evidence ; that is, without evidence, in the proper sense of Has nothing the word. It has nothing to do with the original intuitive pro- '^^t^' °^' ultimate premises of our knowledge; positions, nor ^yj^]^ their uumber or nature, the mode in which with original data; they are obtained, or the tests by which they are distinguished. But, so far as our knowledge is founded on truths made such by evidence, but supplies all tests for that is, derived from facts or other truths pre- propositions. viously kuown, whcthcr those truths be particu- lar truths, or general propositions, it is the prov- ince of Logic to supply the tests for ascertaining the validity of such evidence, and whether or not a belief founded on it would be well ground- ed. And since by far the greatest portion of The greatest our knowledge, whether of particular or general portion of our . „ . ^ knowledge truths, IS avowcdly matter of inference, nearly comes from ^j^^ wholc, not Only of scicncc, but of human inference. •' conduct, is amenable to the authority of logic. CHAP. II.] INDUCTION. 49 INDTTCTION. § 30. That part of logic which infers truths from facts, is called Induction. Inductive rea- soning is the application of the reasoning pro- induction, reasoning cess to a given number of facts, for the purpose applicable. of determining if what has been ascertained re- specting one or more of the individuals is true of the whole class. Hence, Induction is not induction f 1 /■ 1 I • defined. the mere sum of the facts, but a conclusion Irawn from them. The logic of Induction consists in classina; Logic of . . . luductioD. the facts and stating the inference m such a manner, that the evidence of the inference shall be most manifest. § 31. Induction, as above defined, is a process induction of inference. It proceeds from the known to from the known to the unknown. the unknown ; and any operation involving no inference, any process in which the conclusion is a mere fact, and not a truth, does not fall within the meaning of the term. The conclu- The conclu- sion broader sion must be broader than the premises. The than the . premises. premises are facts : the conclusion must be a truth. Induction, therefore, is a process of general- induction, T • 1 • r 1 -11^ process of ization. It IS that operation oi the mind by generaiizar which we infer that what we know to be true 4 50 LOGIC. [book I. in which in a particular case or cases, will be true in all wo conclude, i • i i i j i r • , • that what is cascs which resemble the lormer in certani as- true under gjnrnable respects. In other words. Induction is particular ° '■ circumstan- the process by which we conclude that what cea will be _ tmeuniver- IS truc of Certain individuals of a class is true of the whole class ; or that what is true at cer- tain times, will be true, under similar circum- stances, at all times. Induction § ^2. luduction always presupposes, not only presupposes ^^^^^ ^^^ ncccssaiT observatious are made with accurate and "^ necessary the iieccssary accuracy, but also that the results observations. r- i i • r • i i of these observations are, so far as practicable, connected together by general descriptions : ena- bling the mind to represent to itself as wholes, whatever phenomena are capable of being so represented. To suppose, however, that nothing more is More is required from the conception than that it should necessary ggj-yg to conucct the observations, would be to than to connect the substitute hypothcsis for theory, and imagina- observations we must tion for proof. The connecting link must be mfer from i • i 71 • • ^ r uiem. some character which reaiLy exists m the lacts themselves, and which would manifest itself therein, if the condition could be realized which our organs of sense require. For example ; Blakewell, a celebrated English cattle-breeder, observed, in a great number of CHAP. II.] INDUCTION. 51 individual beasts, a tendency to fatten readily, Example of and in a great number of others the absence oi ti^e Engusb cattle breeder. this constitution : in every individual of the for- mer description, he observed a certain peculiar make, though they differed widely in size, color, &:c. Those of the latter description differed no less in various points, but agreed in being of a different make from the others. These /ac^5 were How he his data; from which, combining them with the the facts: general principle, that nature is steady and uni- 'Wiyhe off' J iuferred. form in her proceedings, he logically/ drew the conclusion that beasts of the specified make have universally a peculiar tendency to fattening. The principal difficulty in this case consisted in what the diflScuIty in making the observations, and so collating and consisted. combining them as to abstract from each of a multitude of cases, differing widely in many re- spects, the circumstances in which they all agreed. But neither the making of the observa- tions, nor their combination, nor the abstraction, nor the judgment employed in these processes, constituted the induction, though they were all preparatory to it. The Induction consisted in in what the the generalization ; that is, in inferring from all consjgted. the data, that certain circumstances would be found in the whole class. The mind of Newton was led to the universal law, that all bodies attract each other by forces 52 LOGIC, [book I. Newton's inference of the law of universal gravitation. How he observed facts and their connections. The use which he made of exact science. What was the result. varying directly as their masses, and inversely as the squares of their distances, by Induction. He saw an apple falling from the tree : a mere fact ; and asked himself the cause ; that is, if any inference could be drawn from that fact, which should point out an invariable antecedent condi- tion. This led him to note other facts, to prose- cute experiments, to observe the heavenly bodies, until fi'om many facts, and their connections with each other, he arrived at the conclusion, that the motions of the heavenly bodies were gov- erned by general laws, applicable to all matter , that the stone whirled in the sling and the earth rolling forward through space, are governed in their motions by one and the same law. He then brought the exact sciences to his aid, and demonstrated that this law accounted for all the phenomena, and harmonized the results of all ob- servations. Thus, it was ascertained that the laws which regulate the motions of the heav- enly bodies, as they circle the heavens, also guide the feather, as it is wafted along on the passing breeze. The ways of ascertaining facts are known: § 33. We have already indicated the ways in which the facts are ascertained from which the inferences * are drawn. But when an inference can be drawn ; how many facts must enter into CHAP. II.] INDUCTION, 53 the premises ; what then' exact nature must be ; and what their relations to each other, and to the inferences which flow from them ; are ques- tions which do not admit of definite answers. Although no general law has yet been discov- ered connecting all facts with truths, yet all the uniformities which exist in the succession of phe- nomena, and most of those which prevail in their coexistence, are either themselves laws of cau- sation or consequences resulting and corollaries capable of being deduced from, such laws. It being the main business of Induction to deter- mine the effects of every cause, and the causes of all effects, if we had for all such processes general and certain laws, we could determine, in all cases, what causes are correctly assigned to what eflects, and v/hat effects to what causes, and we should thus be virtually acquainted with the whole course of nature. So far, then, as we can trace, with certainty, the connection be- tween cause and effect, or between effects and their causes, to that extent Induction is a sci- ence. When this cannot be done, the conclu- sions must be, to some extent, conjectural. but wo do not know certainly, in all cases, ■when we can draw on inference. No general law. Business of Induction. What is necessary. How far a science. 54 LOGIC. [book I. CHAPTER III. DEDUCTION — ^'A1^;RE OF THE SYLLOGISM ITS USES AND APPLICATIONS. DEDUCTION. § 34. We have seen that all processes of Inductive Reasoning, in which the premises are particular '!!!!!!!fnJ' facts, and the conclusions general truths, are called Inductions. All processes of Reasoning, in which the premises are general truths and the Deductive conclusions particular truths, are called Deduc- processes. ^j^^^g^ Hcuce, a deduction is the process of Deduction reasoning by which a particular truth is inferred from other truths which are known or admitted. The formula for all deductions is found in the Syllogism, the parts, nature, and uses of which we shall now proceed to explain. PROPOSITIONS. Proposition, § ^^- ^ proposition is a judgment expressed ludgmentm £^ wovds. Hcnce, a proposition is defined Jogi- words: ^ ' " cally, " A sentence indicative :" affirming or * Section 30. Deductive formula. CHAP. III. PROPOSITIONS. 55 denying; therefore, it must not be ambiguous, must not be ambiguous; for that which has more than one meanmg is norimper- in reaUty several propositions ; nor imperfect, ^fj^^^^^^ nor ungrammatical, for such expressions have no meaning at all. § 36. Whatever can be an object of belief, or even of disbelief, must, when put into words, a proposition • • All 1 ] explained. assume the lorm oi a proposition. All truth and all error lie in propositions. What we call a truth, is simply a true proposition ; and errors its nature,- are false propositions. To know the import of all propositions, would be to know all questions which can be raised, and all matters which are Embraces aii truth and all susceptible of being either believed or disbe- error. lieved. Since, then, the objects of all belief and all inquiry express themselves in propositions, a sufficient scrutiny of propositions and their va- An examina- tion of rieties will apprize us of what questions mankind propositions have actually asked themselves, and what, in the questtoMMU all knowl- actually thought they had grounds to believe. nature of answers to those questions, they have § 37. The first glance at a proposition shows Apropositioc , . . f , , . , is formed b) that it is lormed by putting together two names, p^ting t„o Thus, in the proposition, " Gold is yellow," the "^"^^ ^ ^ •' together. property yellow is affirmed of the substance gold. In the proposition, " Franklin was not born in 56 LOGIC. [book I. England," the fact expressed by the words horn in England is denied of the man Frankhn. A proposition has three parts: Subject, Predicate, and Copula. Subject defined. Copula must be IS or X3 NOT. All verbs resolvable Into " to be." § 38. Every proposition consists of three parts : the Subject, the Predicate, and the Co- pula. The subject is the name denoting the person or thing of which something is affirmed or denied : the predicate is that which is affirm- ed or denied of the subject ; and these two are called the terms (or extremes), because, logically, the subject is placed ^rs^, and the predicate last. The copula, in the middle, indicates the act ot judgment, and is the sign denoting that there is an affirmation or denial. Thus, in the proposi- tion, " The earth is round ;" the subject is the words " the earth," being that of which some- thing is affirmed : the predicate, is the word round, which denotes the quality affirmed, or (as the phrase is) predicated : the word is, which serves as a connecting mark between the subject and the predicate, to show that one of them is af- firmed of the other, is called the Copula. The copula must be either is, or is not, the substan- tive verb being the only vej'h recognised by Logic. All other verbs are resolvable, by means of the verb " to be," and a participle or adjective. For example : •' The Romans conquered :" CHAP. III.] SYLLOGISM. 57 the word " conquered" is both copula and predi- Examples , . . , • . • )} of the cate, being equivalent to " were victorious. copuia. Hence, we might write, " The Romans were victorious," in which were is the copula, and victorious the predicate. § 39. A proposition being a portion of dis- Apropositiou is either course, in which something is affirmed or denied affirmative of .something,, all propositions may be divided into affirmative and negative. An affirmative proposition is that in which the predicate is af- firmed of the subject ; as, " Caesar is dead." A negative proposition is that in which the predicate is denied of the subject ; as, " Caesar is not dead." The copula, in this last species of proposition, in the last, f, , , ,,,.,., the copula ia. consists 01 the words " is not, which is the jg ^^^ sign of negation ; " is" being the sign of affirm- ation. SYLLOGISM. § 40. A syllogism is a form of stating the con- a syllogism consists of nection which may exist, for the purpose of three propo reasoning, between three propositions. Hence, to a leffitimate svllosism, it is essential that ^ . t> ' Two are there should be three, and only three, proposi- admitted: 58 LOGIC. [book 1. and the fhii'd tioiis. Of these, two are admitted to be true, from them. ^^^^ ^^'^ Called the premises : the third is proved from these two, and is called the conclusion. For example : Exomple. Rrajor Term " All tyrants are detestable : Caesar was a tyrant ; Therefore, Caesar was detestable." Now, if the first two propositions be admitted, the third, or conclusion, necessarily follows from them, and it is proved that Caesar was detestable. Of the two terms of the conclusion, the Predi- dcfined, ^^j-g (detestable) is called the m.ajor term, and the Subject (Caesar) the ?ninor term ; and these two terms, together with the term " tyrant," make up the three propositions of the syllogism, Minor Term. — each term being used twice. Hence, every syllogism has three, and only three, different terms. Major The premise, into which the Predicate of the Premise defined, couclusion eutcrs, is called the major premise ; Minor ^^^® othcr is Called the minor premise, and con- Premiae. t^ius the Subjcct of the conclusion ; and the other term, coinmon to the two premises, and with which both the terms of the conclusion were separately compared, before they were compared MiddieTenn. -with cach Other, is Called the middle term. In the syllogism above, "detestable" (in the con- CHAP. III.] SYLLOGISM. 59 elusion) is the major term, and " Caesar" the mi- Example, , pointing out nor term : hence, i^jaj^j. premise, " All tyrants are detestable, niin^r . premise, and IS the major premise, and MiddieTorm. " Caesar was a tyrant," the minor premise, and " tyrant" the middle term. § 41. The syllogism, therefore, is a mere for- ^ J & ' ' Syllogism, mula for ascertaining what may, or what may a mere formula. not, be predicated of a subject. It accomplishes this end by means of two propositions, viz. by comparing the given predicate of the first (a Howappiied. Major Premise), and the given subject of the second (a Minor Premise), respectively with one and the same third term (called the middle term), and thus — under certain conditions, or laws of the syllogism — to be hereafter stated — eliciting the truth (conclusion) that the given predicate must be predicated of that subject. It will be use of the Major seen that the Major Premise always declares, premise. in a general way, such a relation between the Major Term and the Middle Term ; and the Mi- of the Minor nor Premise declares, in a more particular way, such a relation between the Minor Term and the Middle Term, as that, in the Conclusion, ortho 1 1 -HT • Middle Tfc the Minor lerm must be put under tlie Major Term ; or in other words, that the Major Term must be predicated of the Minor Term. 60 LOGIC. [book 1. ANALYTICAL OUTLINE OF DEDUCTION. Reasoning § 42. In evGiy instaiicG in which we reason, in the strict sense of the word, that is, make use of arguments, whether for the sake of refuting an adversary, or of conveying instruction, or of satisfying our own minds on any point, whatever may be the subject we are engaged on, a certain process takes place in the mind, which is one The process, and the same in all cases (provided it be cor- thesame. Tcctly conductcd)," whether we use the inductive process or the deductive formulas. Of course it cannot be supposed that every Everyone ouc is eveii couscious of this proccss in his own not conscious ■ ■, ii • i^x i-^i of the mind; much less, is competent to explain the process, principles on which it proceeds. This indeed is, The same for and cauuot but be, the case with every other every other ,. , . , . i i process, pi'occss respecting which any system has been formed ; the practice not only may exist inde- pendently of the theory, but must have preceded the theory. There must have been Language Eiementsimd bcforc a systeiii of Grammar could be devised ; knowledge of ^^^ musical compositious, previous to the sci- elements, '■ ^ must precede encc of Music. This, by the way, serves to ex- gcneraliza- tion and pose the futility of the popular objection against of principles. Logic J viz. that mcii may reason very well who know nothing of it. The parallel instances ad- duced show that such an objection may be urged CHAP. HI.] ANALYTICAL OUTLINE. 61 in many other cases, where its absurdity would j^jgic be obvious ; and that there is no ground for de- ciding thence, either that the system has no ten- dency to improve practice, or that even if it had not, it might not still be a dignified and inter- esting pursuit. § 43. One of the chief impediments to the lameness of the reasoning attainment of a just view of the nature and ob- process ject of Logic, is the not fully understanding, or kept in mind, not sufficiently keeping in mind the sameness of the reasoning process in all cases. If, as the ordinary mode of speaking would seem to indi- cate, mathematical reasoning, and theological, aii kinds of J ,j •! 1 Ti-ic reasoning are and metaphysical, and pohtical, otc, were essen- alike in tially different from each other, that is, difierent p"^<='p^« kinds of reasoning, it would follow, that suppo- sing there could be at all any such science as we have described Logic, there must be so many different species or at least different branches of Logic. And such is perhaps the most pre- vailing notion. Nor is this much to be won- iJeasonof ,1 . . . . , n 1 '^6 prevaili dered at ; since it is evident to ail, that some men converse and write, in an argumentative way, very justly on one subject, and very erro- neously on another, in which again others excel, who fail in the former. This error may be at once illustrated and re- ing errors 62 LOGIC. [book I. The reasonof moved, by Considering the parallel instance of iuustrated Arithmetic ; in which every one is aware that by example, |.j^g process of a Calculation is not affected bv which shows '■ •' that the rea- the nature of the objects whose numbers are soniug process is before US ; but that, for example, the multipli- always the , • r i • i gjjmg_ cation 01 a number is the very same operation, whether it be a number of men, of miles, or of pounds ; though, nevertheless, persons may per- haps be found who are accurate in the results of their calculations relative to natural philoso- phy, and incorrect in those of political econo- my, from their different degrees of skill in the subjects of these two sciences ; not surely be- cause there are different arts of arithmetic ap plicable to each of these respectively. § 44. Others again, who are aware that the s3oraeview simple systcm of Logic may be applied to all Logic as a peciuiar subjccts whatcvcr, are yet disposed to view it method of i i /- • i reasoning: ^s a peculiar method 01 reasoning, and not, as it is, a method of unfolding and analyzing our reasoning : whence many have been led to talk of comparing Syllogistic reasoning with Moral reasoning; taking it for granted that it is pos- sible to reason correctly without reasoning logi- it is the only cally ; which is, in fact, as great a blunder as if method of . , - reasoning ^^7 ^ue wcrc to mistake grammar lor a pecu- correctiy: y\qx language, and to suppose it possible to speak CHAP. III.] ANALYTICAL OUTLINE. 63 correctly without speaking grammatically. They have, in short, considered Logic as an art of rea- soning ; whereas (so far as it is an art) it is the art of reasonina;: the logician's obiect beinsr, not RiaysdowR "=>' => •' ^ rules, not to lay down principles by which one may reason, which maij, but which but by which all must reason, even though they must be are not distinctly aware of them : — to lay down rules, not which may be followed with advan- tage, but which cannot possibly be departed from in sound reasoning. These misapprehen- Misappre- hensions and sions and objections being such as lie on the objections very threshold of the subject, it would have been hardly possible, without noticing them, to con- vey any just notion of the nature and design of the logical system. § 45. Supposing it then to have been per- operation of , , , . ~ ... 1, reasoning ceived that the operation oi reasonmg is m all should be cases the same, the analysis of that operation '^'^^y^'"^^ could not fail to strike the mind as an interesting matter of inquiry. And moreover, since (appa- rent) arguments, which are unsound and incon- clusive, are so often employed, either from error Because such T . 1 . , , , analysis is or desiffn ; and since even those who are not „„ ,„ o ' necessary to misled by these fallacies, are so often at a loss furnish the to detect and expose them in a manner satis- factory to others, or even to themselves ; it could not but appear desirable to lay down some gen- 64 LOGIC. [book 1, rules for the crul rulcs of reasoning, applicable to all cases ; ion o 1 -which a person misht be enabled the more error and the -J i o discovery of readily and clearly to state the grounds of his truth, own conviction, or of his objection to the argu- ments of an opponent; instead of arguing at random, without any fixed and acknowledged principles to guide his procedure. Such rules Such rules ^yQyifj \yQ analogous to those of Arithmetic, which iu-e analogous " totheruiesof obviatc the tediousness and uncertainty of cal- Arithmetic. culations in the head ; wherein, after much labor, different persons might arrive at different results, without any of them being able distinctly to point out the error of the rest. A system of such rules, it is obvious, must, instead of deserv- They bring iug to be Called the art of wrangling, be more the parties, in . i i • i i r ■ i argument, to Justly characterized as the "art of cuttmg short *"'^*"®" wrangling," by bringing the parties to issue at once, if not to agreement; and thus saving a waste of ingenuity. Every con- § 4g Jn pursuing the supposed investigation, elusion is _ , deducedfrom it will be found that in all deductive processes two proposi- Qy^YY conclusion is deduced, in reality, from two tions, called - '' Premises, other propositious (thence called Premises) ; for ifoneprem- though One of thcsc may be, and commonly is, ise is sup- " '' . •' pressed, it is supprcsscd, it must nevertheless be understood nevertheless understood, as admitted ; as may easily be made evident by supposing the denial of the suppressed premise; CHAP. fTI.] ANALYTICAL OUTLINE. 65 which will at once invalidate the argument. For example ; in the following syllogism : " Whatever exhibits marks of design had an intelligent author: The world exhibits marks of design ; Therefore, the world had an intelligent author :" if any one from perceiving that " the world ex- hibits marks of design," infers that "it must have and is had an intelligent author," though he may not be "^'^'^^^^ *« ° ' & J the argu- aware in his own mind of the existence of any ment, though oue may no! other premise, he will readily understand, if it be be aware denied that " whatever exhibits marks of design must have had an intelligent author," that the affirmative of that proposition is necessary to the validitv of the argument. § 47. When one of the premises is suppressed Enthymemc-. a syllogism (which for brevity's sake it usually is), the argu- with one ment is called an Enthymeme. For example : of it. premise suppressod- " The world exhibits marks of design, Tlierefore the world had an intelligent author," is an Enthymeme. And it may be worth while to remark, that, when the argument is in this objections . made to the state, the objections of an opponent are (or rather assertion or appear to be) of two kinds, viz. either objections ^^^l^Zllt- to the assertion itself, or objections to its force ™'^"'- as an argument. For example : in the above Exampiu instance, an atheist may be conceived either de- 5 66 LOGIC. [book 1. nyino; that the world does exhibit marks of de- Bolh prein- J O ises must be gigrn Or denying that it follows from thence that true, if the /= . argument is it had an intelhgent author. Now it is impor- sound : tant to keep in mind that the only difference in the two cases is, that in the one the expressed premise is denied, in the other the suppressed ; and when for the /orce as an argument of either premise theconciu-' depends on the other premise : if both be admit- sion follows, ^g^^ ^j^g conclusion legitimately connected with them cannot be denied. § 48. It is evidently immaterial to the argu- ment whether the conclusion be placed first or Premise last ; but it may be proper to remark, that a the conciu- premise placed after its conclusion is called the Dion is called ji^f^gQ^j ^f [^ ^ud is iutroduccd by one of those the Reason. •' conjunctions which are called causal, viz. " since," "because," &c., which may indeed be employed to designate a premise, whether it come first or Illative l^st. The iUativc conjunctions " therefore," &c., conjunction, ^^ggig^ate the couclusion. It is a circumstance which often occasions Causes of qytoy and perplexity, that both these classes of error and perplexity, conjunctions have also another signification, be- ing employed to denote, respectively, Cause and Effect, as well as Premise and Conclusion. For Different "^ significations example : if I say, " this ground is rich, because of the . n • ^ ■ ;> ccniunctions. the trccs on it are iiourishmg ; or, " the trees are CHAP. in. J ANALYTICAL OUTLINE. 67 flourishing, and therefore the soil must be rich ;" Examples I employ these conjunctions to denote the con- conjimctiona nection of Premise and Conclusion ; for it is ^'^^ "^'"'^ logically. plain that the luxuriance of the trees is not the cause of the soil's fertility, but only the cause of my knowing it. If again I say, " the trees flourish, because the ground is rich ;" or " the ground is rich, and therefore the trees flourish,' Examples T • 1 ■ , • X 1 J where they 1 am usmg the very same conjunctions to denote denote caust the connection of cause and effect; for in this ^"^^^> to lay down some regular lorm to which necessary, every valid argument may be reduced, and to devise a rule which shall show the validity of every argument in that form, and consequently the unsoundness of any apparent argument which cannot be reduced to it. For example ; if such an argument as this be proposed : Example of " Every rational agent is accountable : an imperfect Brutes are not rational agents ; argument. Therefore they are not accountable , or again : 2d Example. " AH wise legislators suit their laws to the genius of their nation ; Solon did this; therefore he was a wise legislator :" Difficulty of there are some, perhaps, who would not per- detectmgthe j,gj^g ^^y. f^Hacv in such arguments, especiallv if enveloped in a cloud of words ; and still more, when the conclusion is true, or (which comes to the same point) if they are disposed to believe it ; and others might perceive indeed, but might CHAP. III.] ANALYTICAL OUTLINE. 69 be at a loss to explain, the fallacy. Now these To what (apparent) arguments exactly correspond, re- renf'^'*' arguments correspond. spectiveiy, with the following, the absurdity of the conclusions from which is manifest : " Every horse is an animal : A similar Sheep are not horses ; ^^'^'^P'^ Therefore, they are not animals." And " All vegetables grow ; SM similar An animal grows ; example. Therefore, it is a vegetable." These last examples, I have said, correspond Tiieseiast exactly (considered as arguments) with the for- ^ith^the mer ; the question respecting the vahdity of an former, argument being, not whether the conclusion be true, hut whether it follows from the premises adduced. This mode of exposing a fallacy, by This mode o/ bringing forward a similar one whose conclusion , ^^p"*'"^ o o fallacy some- is obviously absurd, is often, and very ad van- *™^^ resorted to. tageously, resorted to in addressing those who are ignorant of Logical rules ; but to lay down such rules, and employ tham as a test, is evi- Toiaydown dently a safer and more compendious, as well best way as a more philosophical mode of proceeding. To attain these, it would plainly be necessary to analyze some clear and valid arguments, and to observe in what their conclusiveness consists. 70 LOGIC. [book I, § 51. Let us suppose, then, such an examin- ation to be made of the syllogism above men- tioned : Example of " Whatever exhibits marks of design had an intelligent author; a perfect rpj^^ world exhibits marks of design ; Therefore, the world had an intelligent author." What is In the first of these premises we find it as- the first sumed universally of the class of " things which premise, exhibit marks of design," that they had an intel- in the second ligent author ; and in the other premise, "the world" is referred to that class as comprehended What we jj^ {[ . j-^qw it is evident that whatever is said of may iufer. the whole of a class, may be said of any thing comprehended in that class ; so that we are thus authorized to say of the world, that " it had an intelligent author." Syllogism Again, if we examine a syllogism with a with a . negative negative conclusion, as, lor example, conclusion. " Nothing which exhibits marks of design could liave been produced by chance ; The world exhibits, &c. ; Therefore, the world could not have been produced by chance," The process the proccss of reasoning will be found to be the of reasoning the same. Same; sincc it is evident that whatever is denied universally of any class may be denied of any thing that is comprehended in that class. CHAP. III. I ANALYTICAL OUTLINE. 71 § 52. On further examination, it will be found mi valid arguments that all valid arguments whatever, which are reducible to based on admitted premises, may be easily re- ' ^^orm"^ duced to such a form as that of the foregoing syllogisms ; and that consequently the principle on which they are constructed is that of the for- mula of the syllogism. So elliptical, indeed, is the ordinary mode of expression, even of those who ordinary •' ^ mode of are considered as prolix writers, that is, so much expressing arguments is implied and left to be understood in the course elliptical. of argument, in comparison of what is actually stated (most men being impatient even, to excess, of any appearance of unnecessary and tedious formality of statement), that a single sentence will often be found, though perhaps considered as a single argument, to contain, compressed into a short compass, a chain of several distinct arguments. But if each of these be fully devel- ^"^ *''•'" o •' fully devel- oped, and the whole of what the author intended oped, they may all be to imply be stated expressly, it will be found that educed into all the steps, even of the longest and most com- ^ f^^m""*' plex train of reasoning, may be reduced into the above form. § 53. It is a mistake to imagine that Aristotle and other logicians meant to propose that this '''*° "^ o i i not mean prolix form of unfolding arguments should uni- that every argument versally supersede, in argumentative discourses, should be 72 LOGIC. [book 1. thrown into the common forms of expression ; and that " to syUogLm.'^ reason logically," means, to state all arguments at full length in the syllogistic form ; and Aris- totle has even been charged with inconsistency for not doing so. It has been said that he " ar- gues like a rational creature, and never attempts That form is to bring his own system into practice." As well ™o7tmth'^' might a chemist be charged with inconsistency for making use of any of the compound sub- stances that ai'e commonly employed, without previously analyzing and resolving them into Analogy to their simplc elements; as well might it be im- the chemist. agined that, to speak grammatically, means, to parse every sentence we utter. The chemist (to pursue the illustration) keeps by him his tests and his method of analysis, to be employed when The analogy any substancc is offered to his notice, the com- position of which has not been ascertained, ov in which adulteration is suspected. Now a fal- To what a lacy may aptly be compared to some adulterated fallacy may , . . ^ . . . , becomp:aod. compouud ; "it cousists 01 an ingenious mixture of tiuth and falsehood, so entangled, so intimate- ly blended, that the falsehood is (in the chemical phrase) held in solution : one drop of sound logic iiow detect- ^^ ^'^^^ ^^^^ which immediately disunites them, ""^ makes the foreign substance visible, and precipi- tates it to the bottom." CHAP. III.] ANALYTICAL OUTLINE. 73 ARISTOTLES DICTUM. § 54. But to resume the investigation of the Form of principles of reasonino; : the maxim resulting from ^""^^^ '^. i^ I O & argument. the examination of a syllogism in the foregoing form, and of the application of which, every valid deduction is in reality an instance, is this : " That whatever is predicated (that is, affirmed Aristotle's or denied) universalis/, of any class of things, may be predicated, in like manner (viz. affirmed or denied), of any thing comprehended in that class." This is the principle commonly called the die- what the tu?Ji de omni et nullo, for the indication of i"''"'^'p'^ IS called. which we are indebted to Aristotle, and which is the keystone of his whole logical system. It is remarkable that some, otherwise iudicious „^ , ., ' J vMiat writers writers, should have been so carried away by h^^vesaidof this princi- their zeal against that philosopher, as to speak pie; and why. with scorn and ridicule of this principle, on account of its obviousness and simplicitv ', ^. ,. . i •' Simplicity a though they would probably perceive at once *'^®'°'' science. in any otlier case, that it is the greatest tri- umph of philosophy to refer many,- and seem- ingly very various phenomena to one, or a very few, simple principles ; and that the more simple and evident such a principle is, provided it be truly apphcable to all the cases in question, the 74 LOGJC. [book I. No solid Ob- greater is its value and scientific beauty. If, jection to the . , - . . , , , , principle Indeed, any prniciple be regarded as not thus ap- ever urged, pjjcable, that is an objection to it of a different kind. Such an objection against Aristotle's dic- tum, no one has ever attempted to establish by been taken ^^^7 ^^^^^ ^^ proof ; but it has oftcn been taken for granted. Jqt granted ; it being (as has been stated) very syuogism commonly supposed, without examination, that not a distinct kind of ar- the syllogism is a distinct kind of argument, and ^"Tform^"' that the rules of it accordingly do not apply, nor applicable to -were intended to apply, to all reasoning what- 3.J1 CflSCS* ever, whei'e the premises are granted or known. Objection: § 55. One objection against the dictum of Aris- t attiosyi- ^Q^T^Q j|. jyiay be worth while to notice briefly, for logisin was •' •' ' intended to i\-^q. g^ke of Setting in a clearer light the real make a dem- onstration character and object of that principle. The ap- plainer: plication of the principle being, as has been seen, to a regular and conclusive syllogism, it has been urged that the dictum was intended to prove and make evident the conclusiveness of such a syllogism ; and that it is unphilo- sophical to attempt giving a demonstration of a demonstration. And certainly the charge to increase would be just, if wc could imagine the logi- tho certainty . , , . -. , ofa cian s object to be, to increase the certainty conclusion. of a conclusion, which we are supposed to have already arrived at by the clearest possible mode CHAP. III.] ANALYTICAL OUTLINE. 75 of proof. But it is very strange that such an This view u 111 entirely idea should ever have occurred to one who had enoneous. even the shghtest tincture of natural philosophy ; for it might as well be imagined that a natural illustration. philosopher's or a chemist's design is to strength- en the testimony of our senses by a priori rea- soning, and to convince us that a stone when thrown will fall to the ground, and that gunpow- der will explode when fired ; because they show according to their principles those phenom.ena must take place as they do. But it v/ould be reckoned a mark of the grossest ignorance and ,,.,.. The object ia stupidity not to be aware that tnen* object is not to prove, not to prove the existence of an individual " °^^' J count for phenomenon, which our eyes have witnessed, but (as the phrase is) to account for it ; that is, to show according to what principle it takes place ; to refer, in short, the individual case to a ffeneral laio of nature. The object of Aris- I'^e object of ° -^ the Dictum totle's dictum is precisely analogous: he had, to point out the general doubtless, no thought of adding to the force of process \o ,- • T 1 11 • 1 • 1 • X -J. which each any individual syllogism ; his design was to point case con- forms. out the general -principle on which that process is conducted which takes place in each syllo- gism. And as the Laws of nature (as they are Laws of nature, gen- called) are in reality merely generalized facts, of eraiized facta which all the phenomena coming under them are particular instances ; so, the proof drawn from 76 LOGIC. [book 1 The Dictum Aristotle's dictum is not a distinct demonstration form of all brought to confirm another demonstration, but is demonsLra- j^g;^-g]y ^ generalized and abstract statement of (ion. "^ " all demonstration whatever ; and is, therefore, in fact, the very demonstration which, under proper suppositions, accommodates itself to the various subject-matters, and which is actually employed in each particular case. How to trace § 56. In oi'dcr to trace more distinctly the the abstract- r i i • i ingand diiierent steps ot the abstracting process, oy reasoning -^y^ich any particular argument may be brought process. ./ r c J o into the most general form, we may first take a syllogism, that is, an argument stated accurately Anarg-ument and at fuU length, such as the example formerly stated at full length, given : " Whatever exhibits marks of design had an intelligent author; The world exhibits marks of design ; Therefore, the world had an intelligent author :" Propositions ^^^ thcii somcwhat generalize the expression, by expressed by substituting (as in Algebra) arbitrary unmean- abstract & \ O / .; terms. jjjg symbols for the significant terms that were originally used. The syllogism will then stand thus : " Every B is A ; C is B ; therefore C is A." Tiie reason- The reasoning, when thus stated, is no less evi- "vaiirt, dently valid, whatever terms A, B, and C respect- CHAP. III.] ANALYTICAL OUTLINE. 77 ively may be supposed to stand for ; such terms and may indeed be inserted as to make all or some general. of the assertions false ; but it will still be no less impossible for any one who admits the truth of the premises, in an argument thus constructed, to deny the conclusion ; and this it is that con- stitutes the conclusiveness of an argument. Viewing, then, the syllogism thus expressed, syiiogismso viewed, it appears clearly that " A stands for any thing affirms gen- whatevcr that is affirmed of a certain entire class" jjetween the (viz. of every B), "which class comprehends or '^™^" contains in it something else" viz. C (of which B is, in the second premiss, affirmed) ; and that, consequently, the first term (A) is, in the conclu- sion, predicated of the third (C). § 57. Now, to assert the validity of this pro- Another form , - . , , . of stating the cess now beiore us, is to state the very dictum dictum, we are treating of, with hardly even a verbal alteration, viz. : 1. Any thing whatever, predicated of a whole The three . things Class , implied. 2. Under which class something else is con- tained ; 3. May be predicated of that which is so con- tained. Thesethree members The three members into which the maxim is correspond to the three here distributed, correspond to the three propo- propositions 78 LOGIC. [bock I. sitions of the syllogism to which they are in- tended respectively to apply. Advantage of The advantage of substituting for the terms, ^arburar"^ in a regular syllogism, arbitrar}^, unmeaning sym- symboisfor j^qJ^^ ^^q)^ g^g letters of the alphabet, is much the the terms. same as in geometry : the reasoning itself is then considered, by itself, clearly, and without any risk of our being misled by the truth or falsity of the conclusion ; which is, in fact, accidental and variable ; the essential point being, as far as Connection, ^^^ argument is concerned, the connection be- the essential point of the ^106671 the prcmiscs and the conclusion. We are argument. thus enabled to embrace the general principle of deductive reasoning, and to perceive its applica- bility to an indefinite number of individual cases. That Aristotle, therefore, should have been ac- Aristotie cuscd of making use of these symbols for the right in using these sym- purposc of darkening his demonstrations, and that too by persons not unacquainted with geom- etry and algebra, is truly astonishing. Syllogism § 58. It bclongs, then, exclusively to a syllo- cqually true . whcnab- gism, propcrly so called (that is, a valid argu- erm j^gj-^| g^ grated that its conclusiveness is evidtmt are used. ' from the mere form of the expression), that if letters, or any other unmeaning symbols, be sub- stituted for the several terms, the validity of the argument shall still be evident. Whenever this CHAP. III.] ANALYTICAL OUTLINE. 79 is not the case, the supposed argument is either whennotso, 11 1 • , ■ 1 1 1 1 1 the supposed unsound and sophistical, or else may be reduced arc-ument (without any alteration of its meaning) into the ^^^^"'^'^' syllogistic form ; in which form, the test just mentioned may be applied to it. § 59. What is called an unsound or fallacious Definition of , . an unsound argument, that is, an apparent argument, which argument. is, in reality, none, cannot, of course, be reduced into this form ; but when stated in the form most nearly approaching to this that is possible, its Whenre- , . . , ^ duced to the lallaciousness becomes more evident, from its fonn, the m- nonconformity to the foregoing rule. For ex- '^Xidenr*^ ample : " Whoever is capable of deliberate crime is responsible ; Example. An infant is not capable of deliberate crime ; Therefore, an infant is not responsible." Here the term "responsible" is affirmed uni- Anaiysisof versally of " those capable of deliberate crime ;" ^^*^ °^'^^"' it might, therefore, according to Aristotle's dic- tum, have been affirmed of any thing contained under that class ; but, in the instance before us, nothing is mentioned as contained under that its defective class ; only, the term " infant" is excluded from "''^'^^^ut "' that class ; and though what is affirm^ed of a whole class may be affirmed of any thing that is contained under it, there is no ground for sup- posing that it may be denied of whatever is not 80 LOGIC. [book 1. so contained ; for it is evidently possible that it the argument '^^J ^^ applicable to a whole class and to some- 13 not good, thing else besides. To say, for example, that all trees are vegetables, does not imply that nothing else is a vegetable. Nor, when it is said, that What the j^jj ^^,]^q j^^.g capable of deliberate crime are re- gtateraent impbes. sponsible, does this imply that no others are responsible ; for though this may be very true, What is to it has not been asserted in the premise before us ; be done in i • i ^ • c the analysis and ui the analysis oi an argument, we are to ''^''" , discard all consideration of what misht be as- argumeut. ° serted ; contemplating only what actually is laid down in the premises. It is evident, therefore, i-heone that such an apparent argument as the above abovedidnot j^^^ ^^^^ comply with the rule laid down, nor comply with r j uieruie. g^n be SO Stated as to comply v/ith it, and is consequently invalid. § GO. Again, in this instance : ^jjothe, " Food is necessary to life ; example. Corn is food ; Therefore corn is necessary to life :" In what the t^g term " ncccssary to life" is affirmed of food, argument is defective, but uot universally ; for it is not said of every hind of food +he meaning of the assertion be- ing manifestly that some food is necessary to life : here again, therefore, the rule has not been complied with, since that which has been predi- CHAP. III.] ANALYTICAL OUTLINE. 81 Gated (that is, affirmed or denied), not of the why we whole, but of apart only of a cerlain class, can- cateor^com not be, on that ground, predicated of whatever ^''"' ^^ , o ' I predicated of is contained under that class. fo"''- DISTRIBUTION AND NON-DISTRIBUTION OF TKRMS. § 61. The fallacy in this last case is, what is Fallacy in the last example. usually described in logical language as consist- ing in the " non-distribution of the middle term ;" Non-distribu- tioii of the that is, its not being employed to denote all the middle term, objects to which it is applicable. In order to understand this phrase, it is necessary to observe, that a term is said to be " distributed," when it is taken universally, that is, so as to stand for all its significates ; and consequently "undistribu- ted," when it stands for only a portion of its sig- nificates.* Thus, "all food," or every kind of what d/stri^ tood, are expressions which imply the distribu- tion of the term " food ;" " some food" would Non-distribu- tion. imply its non-distribution. Now, it is plain, that if in each premiss a part only of the middle term is employed, that is, if it be not at all distributed, no conclusion can How the ex- be drawn. Hence, if in the example formerly ample might adduced, it had been merely stated that " some- ^^^.^^^ * Section 15. 6 82 LOGIC. [book I. thing" (not " whatever," or " every thing") " which exhibits marks of design, is the work of an intelhgent author," it would not have fol- whiitit lowed, from the world's exhibitino; marks of de- would tlion " haveiiDpiied. sign, that that is the work of an intelligent author Words mark- § 62. It is to be obscrvcd also, that the words ingdislribu- )> i • i 11 i- -i • tionornon- "^11 and " every, which mark the distribution not"iway" ^^ ^ term, and "some," which marks its non- expressed. distribution, are not always expressed : they are frequently understood, and left to be supplied by the context ; as, for example, " food is neces- sary ;" viz. " some food ;" " man is mortal ;" viz. Such propo " every man." Propositions thus expressed are sitious ai'e cniied called by logicians " indefinite" because it is left Indefinite. , • 1 i i f r ^ undetermined by the lorm oi the expression whether the subject be distributed or not. Nev- ertheless it is plain that in every proposition the subject either is or is not meant to be dis- tributed, though it be not declared whether But every it is or iiot ; Consequently, every proposition, must be whether expressed indefinitely or not, must be either uudcrstood as either "universal" or "particu- Universal or partir-uiar. lar ;" thosc being called universal, in which the predicate is said of the whole of the subject (or, in other words, where all the significates are included) ; and those particular, in which each. only a part of them is included. For example : CHAP. III.] ANALYTICAL OUTLINE. 83 " All men are sinful," is universal : " some men riiis division are sinful," particular; and this division of prop- / " ositions, having reference to the distribution of the subject, is, in logical language, said to be ac- cording to their " quantUy." § 63. But the distribution or non-distribution Wsfribution of lilt) predi- of the predicate is entirely independent of the catehusno ryfertjncG lo quantity of the proposition ; nor are the signs quantity. " all" and " some" ever affixed to the predicate ; because its distribution depends upon, and is "'is'"«''erenco to quality. indicated by, the " quality' of the proposition ; that is, its being affirmative or negative ; it being a universal rule, that the predicate of a negative proposition is distributed, and of an affirmative, when it is ^ '■ distributed ; undistributed. The reason of this may easily be understood, by considerina; that a term which ^he reason •^ ^ _ of this. stands for a whole class may be applied to (that is, affirmed of) any thing that is comprehended under that class, though the term of which it is xherredicate .^ rr ^ T r I a. j. of aflintialive thus affirmed may be ot much narrower extent j,.op„siaons than that other, and may therefore be far from ""''y ^'^ "P" •^ jjlicablo to coinciding with the whole of it. Thus it may tiie subject, be said with truth, that "the Negroes are unciv- much wider ilized," though the term " uncivilized" be of much ^''*^" ' wider extent than " Negroes," comprehending, besides them, Patagonians, Esquimaux, &c. ; so that it would not be allowable to assert, that 84 LOGIC. [book r. Hence, oniya all who are Uncivilized are Negroes." It is ev- terra is used. i^G'^t, therefore, that it is a pa?^t only of the term "uncivilized" that has been affirmed of " Negroes ;" and the same reasoning applies to every affirmative proposition. But It may It may indeed so happen, that the subject exiLtHh ^^^ predicate coincide, that is, are of equal the subject: g;xtent ; as, for example: "all men are rational animals ;" " all equilateral triangles are equian- gular ;" (it being equally true, that " all rational this not im- animals are men," and that "all equiangular tri- plied in the , ., i jjv i • • • ?• j foi-mofthe angles are equilateral ; ) yet this is not implied expression. ^^ ^j^^ form of the expressiou ; since it would be no less true that " all men are rational ani- mals," even if there were other rational animals besides men. If any part of It is plain, therefore, that if any part of the u^lp'^piiclwr predicate is applicable to the subject, it may be to the sub- affirmed, and of course cannot be denied, of that Joct, it may be affirmed subjcct ; and Consequently, whcu the predicate of the sub- , . . , . ject. is denied of the subject, this implies that no part of that predicate is applicable to that sub- ject ; that is, that the whole of the predicate is Ka predicate denied of the subject: for to say, for example, Li^ecl "he^ that " no beasts of prey ruminate," implies that whole predi- jjgag^g ^f pj^.gy ^re excluded from the whole class cate is i •! denied of of ruminant animals, and consequently that " no the subject. . . „ * i rummant animals are beasts oi prey. And CHAP. III. J ANALYTICAL OUTLINE. 85 hence results the above-mentioned rule, that the Distribution ,. ., . r- 1 T • • T 1 • of predicate distribution oi the predicate is impued in nega- i„,piit,jii, tive propositions, and its non-distribution in af- "«^s"tive '■ '■ propositions: firmativeS. non-Uistribu- tion in aflirmatives. § 64. It is to be remembered, therefore, that Not sufficient for the mid- it is not sufficient for the middle term to occur die term to , . . . • r 1 occur in a in a universal proposition ; since it that propo- universal sition be an affirmative, and the middle term be p'"'^!'"*'"'"'- the predicate of it, it will not be distributed. For example : if in the example formerly given, it had been merely asserted, that " all the works of an intelligent author show marks of design," and that " the universe shows marks of design," u must be so nothing could have been proved ; since, though *^^Xthe both these propositions are universal, the middle *^'''"* °^ '^® conclusion, term is made the predicate in each, and both are that those terras may be affirmative ; and accordingly, the rule of Aris- compared to- totle is not here complied with, since the term ^^ '^'^' " work of an intelligent author," which is to be proved applicable to " the universe," would not have been affirmed of the middle term (" what shows marks of design") under which " universe" is contained ; but the middle term, on the con- trary, would have been affirmed of it. If, however, one of the premises be negative, if oneprem- the middle term may then be made the predicate '^^ ^^^s» 86 LOGIC. [book I. live, the mid- of that, and will thus, according to the above bemadJihl remark, be distributed. For example : predicate of that, and will ,, ,t • . • i i • , ..... " ]\o ruminant animals are predacious : be distnb- '^ uted. The lion is predacious ; Therefore the lion is not ruminant ;" this is a valid syllogism ; and the middle term (predacious) is distributed by being made the The form of predicate of a negative proposition. The form, thissyiio- jj;^(jgg(j Qf the syllogism is not that prescribed gism will not '' ° '■ beiiiatpre- by the dictum of Aristotle, but it may easily be scribed by the dictum, reduced to that form, by stating the first prop- but inny be . . , _ , , . . , reduced to it. osition thus : " J\o prcdacious animals are ru- minant;" which is manifestly implied (as was above remarked) in the assertion that "no ru- minant animals are predacious." The syllogism will thus appear in the form to which the dictum applies. AM argil- §65. It is not every argument, indeed, that °i7rldi|™d* can be reduced to this form by so short and sim- bysoshorta |g an alteration as in the case before us. A process. -i longer and more complex process will often be required, and rules may be laid down to facilitate this process in certain cases ; but there is no sound argument but what can be reduced into But all argu- this form, without at all departing from the real meutsmay ^g^uing and drift of it; and the form will be CHAP. IK.] ANALYTICAL OUTLINE. 87 found (though more prolix than is needed for be reduced ordinary use) the most per argument can be exhibited. , . ^ , . . , . , to the pre- ordmary use) the most perspicuous m which an gcribedform. § 66. All deductive reasoning whatever, then, AUdeauctivc rests on the one simple principle laid down by rests on the Aristotle, that ^^'=''^'"- " What is predicated, either affirmatively or negatively, of a term distributed, may be predi- cated in like manner (that is, affirmatively or neg- atively) of any thing contained under that term." So that, when our object is to prove any prop- what are the osition, that is, to show that one term may rightly p™"^^^^" be affirmed or denied of another, the process which really takes place in our minds is, that we refer that term (of which the other is to be thus predicated) to some class (that is, middle term) of which that other may be affirmed, or denied, as the case may be. Whatever the subject-mat- Thereasoii- ing always ter of an argument may be, the reasoning itself, the same. considered by itself, is in every case the same process; and if the writers against Logic had Mistakes of kept this in mmd, they would have been cautious Logic. of expressing their contempt of what they call " syllogistic reasoning," which embraces all de- ductive reasoning; and instead of ridiculing Aris- totle's principle for its obviousness and simplicity, Anstotie-s would have perceived that these are, in fact, its ''""'"''® 88 LOGIC. [book I. simple and highest praise: the easiest, shortest, and most evident theory, provided it answer the purpose of explanation, being ever the best. RULES FOR EXAMINING SYLLOGISMS. rests of the § 67. The following axioms or canons serve validity of syllogisms, ss tests of the Validity of that class of syllo- gisms which we have considered. 1st test. 1st. If two terms agree with one and the same- third, they agree with each other. ad test. 2d. If one term agrees and another disagrees with one and the same third, these two disagree with each other. The first the On the former of these canons rests the va- teat of all _ affirmative Hdity of affirmative conclusions ; on the latter, conclusions. f. , . c 1 1 • ^ r ^ , The second ^^ negative : lor, no syllogism can be laulty of negative, ^y^jj^j^ ^Qgg j-^qj- yjolate these canons ; none cor- rect which does ; hence, on these two canons are built the following rules or cautions, which are to be observed with respect to syllogisms, for the purpose of ascertaining whether those canons have been strictly observed or not. Every syiio- 1 st. Every syllogism has three and only three three and t^^^s ; viz. the middle term and the two terms only three q|- ^j^g Couclusion t the tcmis of the Conclusion terms. are sometimes called extremes. Every syiio- 2d. Evcry syllogism h is three and only three CHAP. III.] ANALYTICAL OUTLINE, 89 ■propositions; viz. the major premise ; the minor gismhas ■, , , . three and premise ; and the conclusion. ^niy three 3d. If the middle term is ambiguous, there P™P"«'tions. . Middle term are in reality two middle terms, in sense, though must not bo but one in sound. ambiguous. There are two cases of ambiguity: 1st. Where Two cases the middle term is equivocal ; that is, when used istcase. in different senses in the two premises. For example : " Light is contrary to darkness ; Feathers are light ; therefore, Feathers are contrary to darkness." Example. 2d. Where the middle term is not distrib- 2d case. uted ; for as it is then used to stand for a part only of its signijicates, it may happen that one of the extremes is compared with one part of the whole term, and the other with another part of it. For example : Lgain Ebcample " White is a color ; Blark is a color ; therefore, Black is white." " Some animals are beasts ; Some animals are birds ; therefore, Some birds are beasts." The middle 3d. The middle term, therefore, must he dis- term must be once distrib- trihuted, once, at least, in the premises ; that is, uted: 90 LOGIC. [book I. and cnce is Bufficiuut. No term must be dislri bil- led in the conclusion which was Dot distribu- ted in a premise. Examp'.c. Negalivft premises prove noth- Esaniple. by being the subject of a universal,* or predi- cate of a negative ;t and once is sufficient ; since if one extreme has been conapared with a part of the middle term, and another to the whole of it, they must have been compared with the same. 4th. No term must he distributed in the con- clusion which was not distributed in one of the premises; for, that would be to employ the whole of a tei'm in the conclusion, when you had employed only a part of it in the premise ; thus, in reality, to introduce a fourth term. This is called an illicit process either of the major or minor term. J For example : " All quadrupeds are animals, A bird is not a quadruped ; therefore, It is not an animal." Illicit process of the major. 5th. From negative premises you can infer nothing. For, in them the Middle is pronounced to disagree with both extremes ; therefore they cannot be compared together : for, the extremes can only be compared when the middle agrees with both ; or, agrees with one, and disagrees with the other. For example : " A fish is not a quadruped ;" " A bird is not a quadruped," proves nothing'. * Section 62. f Section 63. X Section 40, III.] ANALYTICAL OUTLINE. 91 6lh. If one premise ba negative, the conclu- ifoneprem- , . f, . , . , ise is nega- 5/071 must be negative; tor, in that premise the tjve, the middle term is pronounced to disagree with one '=""'^'''*'"" ^ c) •^Yill be iiegar of the extremes, and in the other premise (which ''^'®; of course is affirmative by the preceding rule), to agree with thi other extreme ; therefore, the extremes disagreeing with each other, the con- elusion is negative. In the same manner it may andrecipro be shown, that to prove a negative conclusion, one of the premises must be a negative. By these six rules all Syllogisms are to be what fol- lows from tried; and from them it will be evident, 1st, these six that nothing can be proved from two particular premises ; (since you will then have either the middle term undistributed, or an illicit process. For example : " Some animals are sagacious ; Some b?asts are not sagacious ; Some beasts are not animals.") And, for the same reason, 2dly, that if one of ea inferenca the premises be particular, the conclusion must be particular. For example : " All who fight bravely deserve reward ; " Some soldiers fight bravely ;" you can only infer that " Some soldiers deserve reward :" for to infer a universal conclusion would be an illicit process of the minor. But from two Example. 92 LOGIC. [b( rwouniver- uiiivei'sal Premises you cannot always infer a sal premises . i /-i i • t^ i doaotaiways universal Conclusion. For example: give a uni- versal con- " All gold is precious ; <=i"^i°'i' All gold is a mineral ; therefore, Some mineral is precious.' I And even when we can infer a universal, we are always at lihei'ty to infer a particular ; since what is predicated of all may of course be pre dicated of some. OF FALLACIES. Definition of § 68. By a fallacy is commonly understood afaUacy. ^ r ■ i • i " any unsound mode oi arguing, which appears to demand our conviction, and to be decisive of the question in hand, when in fairness it is Detection of, not." In the practical detection of each indi- acuteness. vidual fallacy, much must depend on natural and acquired acuteness ; nor can any rules be given, the mere learning of which will enable us to apply them with mechanical certainty and Hints and rcadiuess ; but still we may give some hints that rules useful, ^^.^j j^^^ ^^ coiTcct general views of the subject, and tend to engender such a . habit of mind, as will lead to critical examinations. Same of Lo- Indeed, the case is the same with respect to gicingenerai. j^^^^^ j^^ general; scarcely any one would, in ordinary practice, state to himself either his CHAP. III.] ANALYTICAL OUTLINE. 93 own or another's reasoning, in syllogisms at full Logic tends , , p.,.. -ii-i ••! to cultivate length ; yet a lamiharity with logical principles ,j^(,its of tends very much (as all feel, who are really well '^i'^™*""' acquainted with them) to beget a habit of clear and sound reasoning. The truth is, in this as in manv other thinsrs, there are processes sfoinsr Thehabu fixed, we on in the mind (when we are practising any naturally foi- thing quite familiar to us), with such rapidity processes. as to leave no trace in the memory ; and we often apply principles which did not, as far as we are conscious, even occur to us at the time. § 69. Let it be remembered, that in every conclusion f, . 1 • 11 II follows from process oi reasoning, logically stated, the con- t^v^antece- clusion is inferred from two antecedent propo- -, if , ... any, either in be any, must be either, the premises 1st. In the premises ; or, 2d. In the conclusion (when it does not follow orconciu- - , , sion, or both, irorn them) ; or, 3d. In both. In every fallacy, the conclusion either does or does not follow from the premises. When the fault is in the premises ; that is, when in the when they are such as ought not to have been p*"®™'^®^' assumed, and the conclusion legitimately follows from them, the fallacy "s called a Material Fal- 94 Logic. [book i. lacy, because it lies in the matter of the argu- ment. When in the Where the conclusion does not follow from conclusion. ^, ... .^ , , „ the premises, it is manifest that the fault is in the reasoning, and in that alone: these, there- fore, are called Logical Fallacies, as being prop- erly violations of those rules of reasoning which it is the province of logic to lay down. When in When the fault lies in both the premises and reasoning, the fallacy is both Material and Logical both. Rules for § 70. In examining a train of argumentation, examining a^ , • -c r i\ i • •. train of iu-- ^^ ascertain it a iallacy have crept into it, the guraent. foUowing poiuts would naturally suggest them- selves : tstRuie. 1st. What is the proposition to be proved? On what facts or truths, as premises, is the ar- gument to rest ? and, What are the marks of truth by which the conclusion may be known ? SdUuie. 2d. Are the premises both true? If facts, are they substantiated by sufficient proofs ? If truths, were they logically inferred, and from correct premises? 3d Rule. 3d. Is the middle term what it should be, and the conclusion logically inferred from the prem- ises ? Suggestions These general suggestions may serve as guides serve as . . . guides, ^^ examining arguments lor the purpose of de- CHAP, in.] ANALYTICAL OUTLINE. 95 tecting fallacies ; but however perfect general to detect rules may be, it is quite certain that error, in its thousand forms, will not always be separated from truth, even by those who most thoroughly understand and carefully apply such rules CONCLUDING REMARKS. § 71. The imperfect and irregular sketch which Logic corresponds has here been attempted of deductive logic, may ^nh the suffice to point out the general drift and purpose ''eason"ib's m ^ "^ '^ '^ Geometry. of the science, and to show its entire correspond- ence with the reasonings m Geometry. The analytical form, which has here been adopted. Analytical is, generally speaking, better suited for introdu- "™' cing any science in the plainest and most inter- esting form ; though the synthetical is the more synthetical regular, and the more compendious form for sto- ring it up in the memory. § 72. It has been a matter about which wri- induction: 1-1 T£r 1 1 .1 I • does it form ters on logic have dinered, whether, and in con- a part of formity to what principles. Induction forms a °^^ part of the science ; Archbishop Whately main- whateiy's taming that logic is only concerned m inierrmg truths from known and admitted premises, and that all reasoninff, whether Inductive or Deduc- live, is shown by analysis to have the syllogism 96 LOGIC. [book I. Mill's views, for its type ; while Mr. Mill, a writer of perhaps greater authority, holds that deductive logic is but the carrying out of what induction begins ; that all reasoning is founded on principles of in- ference ulterior to the syllogism, an.d that the syllogism is the test of deduction only. Without presuming at all to decide defini- tively a question which has been considered and Reasons for passcd upou by two of the most acute minds of the course taken. the age, it may perhaps not be out of place to state the reasons which induced me to adopt the opinions of Mr. Mill in view of the par- tfcular use which I wished to make of logic. reading Ob- § 73. It was, as stated in the general plan, jects of the plan- one of my leading objects to point out the cor- respondence between the science of logic and the science of mathematics : to show, in fact. To show that that mathematical reasoning conforms, in every mathemati- , . , ^ , . ... cai reasoning I'sspcct, to the sti'ictcst rulcs 01 logic, and IS in- conformsto deed but logic applied to the abstract quantities, Jo^ical rules, o i j. Number and Space. In treating of space, about which the science of Geometry is conversant, we shall see that the reasoning rests mainly on the Axioms, how axiouis, and that these are established by induc- eetablished. ,• mu r • i • u tive processes. 1 he processes oi reasoning which relate to numbers, whether the numbers are rep- resented by figures or letters, consist of two parts. CHAP. III.] ANALYTICAL OUTLINE. 97 1st. To obtain formulas for, that is, to express in the language of science, the relations between tlie quantities, facts, truths or principles, what- Two pans of the reasoning ever they may be, that form the subject of the process. reasoning ; and, 2dly. To deduce from these, by processes purely logical, all the truths which are implied in them, as premises. § 74. Before dismissing the subject, it may ah inauo- tion may be be well to remark, that every induction may thrown into be thrown into the form of a syllogism, by sup- JJ-^^g plying the maior premise. If this be done, we ^yi'ogism.by t' J ^ •> t- admitting a shall see that something equivalent to the uni- proper major r • r 1 f Ml premise. jormity oj the course of nature will appear as the ultimate major premise of all inductions ; and will, therefore, stand to all inductions in the relation in which, as has been shown, the major premise of a syllogism always stands to the conclusion ; not contributing at all to prove it, but being a necessary condition of its being proved. This fact sustains the view taken by Mr. Mill, as stated above; for, this ultimate ma- now this . . ^ . . . f. major prera jor premise, or any substitution lor it, is an inter- iseia obtain- ence by Induction, but cannot be arrived at by ^^ means of a syllogism. 7 O <1 [iH o ^ Eh H W p^ O Ed 1— 1 H-l oi CO "* 02 r/) H m ^; W < CO <5 BOOK II. MATHEMATICAL SCIENCE. CHAPTER I. QUANTITY ANB MATHEMATICAt, SCIENCE DEFINED— DIFFERENT KINDS OP QUAN- TITY — LANGUAGE OF MATHKMATICS EXPLAINED — SUBJECTS CLASSIFLED— UNIT OF MEASURE DEFINED — MATHEMATICS A DEDUCTIVE SCIENCE. QUANTITY. § 75. Quantity is any tiling which can be Quantity increased, diminislied, and measnred. § 76. Mathematics is the science of quan- Mathematics tity; that is, the science Avhich treats of the measures of quantities and of their relations to each other. § 77. There are two kinds of quantity; Num- Kinds of 1 T o quantity. ber and bpace. N" U 51 B E K . § 78. A NUMBER is a unit, or a collection Number defined. of units. 100 MATHEMATICAL SCIENCE. [BOOK II. Abstract. AlST ABSTRACT NUMBER is One whose Unit is not named; as, one, two, three, &c. Denominate. A DENOMIITATE NUMBER is One AvllOSe Unit is named; as three feet, three yards, thi'ee pounds. Such numbers are also called concrete numbers. How we Ob- How do Tvc acquire our first notions of num- of number, bers ? Bj first presenting to the mind, through the eye, a single thing, and calling it one. Then presenting two things, and naming them two ; then three tilings, and naming them three; and so on for other numbers. Thus, It is done by ^6 acquirc primarily, in a concrete form, our perception g]gjjjgjj|-g^j.y ^otious of number, by perception, and comparison, and reflection ; for, we must first reflection. -^ ' ' ^ perceive how many things are numbered; then compare what is designated by the word one. Reasons, with what is designated by the words two, three, &o., and then reflect on the results of such comparisons until we clearly apprehend the difference in the signification of the words. Haying thus acquired, in a concrete form, our conceptions of numbers, we can consider num- bers as separated from any particular objects. Two axioms ^^^^ i\\\\^ form a conception of them in the ab- necessary for the forma- stract. We require but two axioms for the tion of num- bers, formation of all numbers : 1st axiom. Ist. That oue may be added to any number, and that the number which results will be CHAP. I.] DEFIlSriTIOivTS. 101 greater by one than the number to which the one was added. 2d. That one may be divided into any num- 2d axiom, ber of equal parts. ^ § 79. Under Xumber, we have four species, or Four kinds of number. subdivisions, each differing from the other three, in the unit of its base : thus, 1. Abstract Number, wlien the base is the ab- Abstract, stract unit one : 3. Number of Currency, Avhen the base is a Currency, unit of currency, as one dollar: 3. Number of Weight, when the base is a unit weight of weight, as one pound : 4. Number of Time, when the base is a unit Time. of time, as one day. Hence, in number, we have four kinds of Four kinds of units, units: Abstract Units ; Units of Currency; Units of Weight : and Units of Time. SPACE. § 80. Space is indefinite extension. We ac- space defined. quire our ideas of it by obserAang that parts of it are occupied by matter or bodies. This enables us to attach a definite idea to the word place. We are then able to say, intelligibly, that a point is that which has place, or posi- a point. 102 MATHEMATICAL SCIEIirCE. [BOOK II. tion in space, Avithout occupying any part of it. Having conceived a second point in space, we can nnderstand the important axiom, "A Axiom con- Straight line is tlie shortest distance between Btrai^'^hfiiue ^^^ points*" and this line we call length, or a dimension of space. § 81. If we conceive a second straight line to be drawn, meeting the first, but lying in a direction directly from it, we shall liave a sec- defliied. ond dimension of space, which we call hreadtli. If these lines be prolonged, in both directions, they will include four portions of space, winch make up what is called a plane surface, or plane : hence, a plane has two dimensions, length and breadth. If now we draAv a line on either side of this plane, we shall have another dimen- sion of space, called thiclcness : hence, space has three dimensions — length, breadth, and thick- A plane deflued. Space has three dimen- sions. Figure defined. Line de- fined. § 83. A portion of space limited by bounda- ries, is called a Figure. If such portion of space have but one dimension, it is called a line, and may be limited by two points, one at each Two kinds extremity. There are two kinds of lines, straight of lines: ^^^^-^ curved. A straight line, is one which does straight and ° curved. j^q|; change its direction between any two of its Difference. CHAP. I.] SPACE. 103 points, and a curved line constantly changes its direction at every point. § 83. A portion of space having two dimen- surface : " sions is called a surface. There are two kinds of surfaces — Plane Surfaces and Curved Sur- cumTd. faces. With the former, a straight line, having two points in common, will always coincide, however it may be placed, while with the latter '' -^ Boundaries it will not. The boundaries of surfaces are of » ^"'f^ce. lines, straight or curved. § 84. A limited portion of space, having three volumes, dimensions, is called a Volume. All volumes Boundaries, are bounded by surfaces, either plane or curved. § 85. An" angle is the amount of divergence of two lines, of two planes, or of several planes, ' ^ ' ^ ' Angles. meeting at a point; and is measured, like other magnitudes, by comparing it with its unit of measure. Hence, in space, we have four units, measure, differing in kind : 1. Linear Units, for the measurement of Linear, lines ; 2. Units of Surface, for the measurement of surface, surfaces ; 3. Units of Volume, for the measurement of volume, volumes ; and 104 MATHEMATICAL SCIEKCE. [BOOK II. Angle. 4. Units of Angles, for the measurement of angles. Eight units. § 86. Besides the eight kinds of units, four of number and four of space, embraced in the above classification, and in which the units of each class are connected by known laws, there are yet isolated denominate numbers, such as five chairs, six horses, seven things, &c., which Unit? with- do uot admit of classification, because they have out law. no law of formation. Neither does this classi- units. fication include the Infinitesimal Units, which are specially treated of in Chapter V., Book II., and which are the elements of a very important branch of Mathematical Science. Language § 87. The language of Mathematics is mixed. mathema- Althougli composcd mainly of symbols, which are defined Avith reference to the uses which are made of them, and therefore have a pre- cise signification; it is also composed, in part, of words transferred from our common lan- SjTnbois miaare. The symbols, although arbitrary signs, general. & ^ J ' o J b > are, nevertheless, entirely general, as signs and instruments of thought; and when the sense in Avhich they are used is once fixed, by definition. Sense un- they preserve throughout the entire analysis changed. precisely the same signification. The meaning CHAP. I.] LANGUAGE OF MATHEMATICS. 105 of the words borrowed from our common vocab- Words bor- ulary is often modified, and sometimes entirely common changed, when the words are transferred to the are mocimed lano-uao-e of science. They are then used in .^ ''"f' ^^ed in a ° ^ '' technical particular sense, and are said to have a technical ^®'^^°- signification. § 88. It is of the first importance that the Lan-uage elements of the language be clearly understood, untostood : — that the signification of every word or sym- bol be distinctly a^iprehended, and that the con- nection between the thought and the word or symbol which expresses it, be so well estab- lished that the one shall immediately suffg-est ,, •^ °° Matliemati- the other. It is not possible to pursue the sub- *="' reason- ings require tie reasonings of Mathematics, and to carry out it- the trains of thought to which they give rise, without entire familiarity with those means which the mind employs to aid its investiga- tions. The child cannot read till he has learned ^ Cannot nse the alphabet ; nor can the scholar feel the deli- ^"^ ^'*°' '- guage well cate beauties of Shakspeare, or be moved by the *''' ^'^ ^ -^ know it. sublimity of Milton, before studying and learn- ing the language in which their immortal thoughts are clothed. § 89. All Quantities, whether abstract or con- Quantities Crete, are, in mathematical science, presented ^'*^ ''"P"'®" 5* 106 MATHEMATICAL SCIElSrCE. [iJOOK II. sentedby to the miud bj arbitrary symbols. Thev are symbols ; _ ^ and are opei-- vieAved and Operated on through these symbols ated on bj'^ . these gym- which represent them; and all operations are indicated by another class of symbols called Signs. signs. These, combined with the symbols whatconsti- ^^^^^^^^ represent the quantities, make np, as i!rf!„' -f ^^'6 hn\e stated above, the pure mathematical lan n ^ ^^^ same. technical Avord, is accurately denned, so that to each there is attached a definite and precise idea. Thus, the language is made so exact and Language exact. certain, as to admit of no ambiguity. II^FIlSriTESIMAL CALCULUS. 8 97. The language of the Infinitesimal Cal- Language ^ ® ° (ifihe cuius is very simple. Its chief element is the inflniteshnai Calculus. letter d, which, when written before a quantity, denotes that that quantity increases or decreases according to the law of continuity, and the ex- pression thus arising is one link in that law. Thns, dx denotes that the quantity represented What does dx denote. by X, changes according to the law of continuity, and that dx is the unit of that change. 110 MATHEMATICAL SCIENCE. [BOOK II. PURE MATHEMATICS. Pure Mathe- § 98. The Pure Mathematics embraces all the matics. principles of the science, and all the processes by which those principles are developed from Number and the abstract quantities, Number and Space. All Space. the definitions and axioms, and all the truths deduced from them, are traceable to these two sources. Mathema- § 99. Mathematics, in its primary significa- tics, as used by the an- tiou, as uscd by the ancients, embraced every acquired science, and was equally applicable to all branches of knowledge. Subsequently it was restricted to those brandies only which Avere acquired by severe study, or discipline, and Embraced its votarics Were called Disciples. Those sub- all subjects . which wore jects, therefore, which required patient mvesti- in their iia- ga^tiou, cxact reasoning, and the aid of the ma- *'^'^' thematical analysis, were called Disciplinal or Mathematical, because of the greater evidence in the arguments, the infallible certainty of the conclusions, and the mental training and de- velopment which srxh exercises produced. Pure Mathe- § 100. The Purc Mathematics is based on definitions and intuitive truths, called axioms, What are its foundations, which are inferred from observation and expe- CHAP. I.] PURE MATHEMATICS. Ill rience ; that is, observation and experience fur- Premises, nish tl]e information necessary to such intui- tive inductions.* From these definitions and axioms, as premises, all the trnths of the science Reasoning, are established by processes of deductive reason- ing; and there is not, in the whole range of its tests of mathematical science any logical test of truth, lut in a conformity of the conclusions to the what they are. definitions and axioms, or to such j^^'i^^dples or propositions as have been established from them. Hence, we see, that the science of Pure Mathe- in what the ,. I'l -i. !••/>• 1 science con- matics, Avhich consists merely m inferiing-, by gj^^g^ fixed rules, all the truths Avhich can be deduced from given premises, is purely a Deductive i« purely Deductive. Science. The precision and accuracy of the definitions ; the certainty which is felt in the truth of the axioms ; the obvious and fixed re- Precision of its language. lation betAveen the sign and the thing signified ; and the certain formulas to which the reason- ing processes are reduced, have given to mathe- Exact, niatics the name of '-Exact Science." ' science. § 101. We have remarked that all the rea- ah rrason- sonings of mathematical science, and all the definitions truths which they establish, are based on the """^ ''^^''"''• definitions and axioms, Avhich correspond to the * Section 37. 112 MATHEMATICAL SCIEXCE. [BOOK II. major premise of the syllogism. If the resem- blance which the minor premise asserts to the Relations middle term were obvions to the senses, as it "'is in the proposition, "Soci-ates was a man," or were at once ascertainable by direct observa-' tion, or were as evident as the intnitiye trnth, "A whole is eqnal to the sum of all its parts;" Deductive there would be no necessity for trains of rea- necessuiy. soning, and Deductive Science wonld not exist. Trains of Trains of reasoning are necessary only for the reasoiimg. ^_^^^ ^^ extending the definitions and axioms to What they otlicr cases in which we not only cannot di- accomp is . j.gg^jy observe what is to be proved, but cannot directly observe even the mark which is to prove it. Syllogism, § 103. Although the syllogism is the ultimate ^^of'lieduc-'''^ test in all deductive reasoning (and indeed in *^°°" all inductive, if Ave admit the uniformity of the course of nature), still w^e do not find it con- venient or necessary, in mathematics, to throw every proposition into the form of a syllogism. . . , The definitions and axioms, and the propo- Axioms and ^ ^ definiii.ms, gitions established from them, are our tests of tests of truth. truth; and whenever any new proposition can be brought to conform to any one of these Apropos!- tests, it is regarded as proved, and declared to tion : when proved, be true. CHAP. I.] MIXED MATHEMATICS. 113 § 103. When general formulas have been When a framed, determining the limits within which maybere- the deductions may be drawn (that is, what ^proved.^ shall be the tests of truth), as often as a new case can be at once seen to come within one of the formulas, the principle applies to the new case, and the business is ended. But new cases Trains of reasoning: are continually arising, which do not obviously whyncces- come within any formula that will settle the '^' questions we want solved in regard to them, and it is necessary to reduce them to such for- mulas. Tliis gives rise to the existence of the They give 1 , . 1 r'*e t^o the science oi mathematics, requiring the highest ecicnceof scientific genius in those who contributed to its ^ ^^^.^^ ' creation, and calling for a most continued and vigorous exertion of intellect, in order to appro- priate it, when created. MIXED MATHEMATICS. § 104. The Mixed Mathematics embraces the Mixed applications of the principles and laws of tlie tics. Pure Mathematics to all investigations in which the mathematical language is employed and to the solution of all questions of a practical na- ture, whether they relate to abstract or concrete quantity. 8 114 MATHEMATICAL SCIENCE. [HOOK II. QUAN'TITT MEASURED. Quantity. §105. Quantity has been defined, "anything which can be increased, diminished, and nieas- increased ^^i-ed." The terms increased or diminished, are and diminished, easily nnderstood, implying merely the property of being made larger or smaller. The term measured is not so easily comprehended, because it has only a relative meaning. Measured. The term "measured," applied to a quantity, implies the existence of some known quantity wiiat it of the same kind, which is regarded as a stand- '^'^''' ard. With this standard, the quantity to be meas- ured is compared with respect to its extent or standard: magnitude. To such standai-d, Avhatever it may is called be, we give the name of iinity, or unit of '^""^' measure; and the number of times Avhich any quantity contains its unit of measure, is the numerical value of the quantity measured. The Magnitude: extent or magnitude of a quantity is, therefore, tivc. merely relative, and hence, we can form no idea of it, except by the aid of comparison. Space, Space: for example, is entirely indefinite, and we meas- Indeflnite. lire parts of it by means of certain standards, Measure- Called measures ; and after any measurement is ment ascer- tains rcia- completed, we have only ascertained the relation or proportion which exists between the stand- ard we adopted and the thing measured. Hence, CHAP. I.] QUAKTITT MEASURED. 115 measurement is, after all, but a mere process a process of „ . compai'isoiL 01 comparison. § 106. The quantities, Number and Space, are but Number and vague and indefinite conceptions, until we compare known by them with their units of measure, and even these *^°'^i'^"^<''*- units are arrived at only by processes of comparison. Comparison is the great means of discovering the comparison relations of things to each other, as well in general method, logic, as in the science of mathematics, which develops the processes by which quantities are compared, and the results of such comparisons. § 107. Besides the classification of quantity Quantity, into Number and Space, we may, if we please, divide it inta Abstract and Concrete. An ab- Abstract, stract quantity is a mere number, in Avhich the unit is not named, and has no relation to mat- ter or to the kind of things numbered. A con- concrete. Crete quantity is a definite object or a collec- tion of such objects. Concrete quantities are expressed by numbers and letters, and also by iiow repre- sented, lines, straight and curved. The number " three" ^ ^ ° Example is entirely abstract, expressing an idea having of the abstract. no connection with things; while the number "three pounds of tea," or "three apples," pre- Example sents to the mind an idea of concrete objects. concrete. So, a portion of space, bounded by a surface, all 116 MATHEMATICAL SCIENCE. [BOOK II. the points of wliicli are equally distant from a Sphere Certain point within called the centre, is but a mental conception of form ; but regarded as a defined, portion of space, gives rise to the additional idea of a named and defined thins:. COMPAEISON" OF QUAIfTITIES. Mathematics § 108. We liaYe Seen that the pure mathe- HithNum- niatics are concerned with the two quantities, Space Number and Space. We have also seen, that rea- Keasoning soning nccessarilj involves comparison : hence, comparison, mathematical reasoning must consist in com- paring the quantities wliich come from Number and Space with each other. Two qiianti- § ^^^' -^^J ^^^ quantities, compared with ties can sns- qq^q]^ other, must necessarilv sustain one of two tain but two ' •' relations, relations: tbey must be equal, or unequal. What axioms or formulas have we for inferring the one or the other? AXIOMS FOR IlfFERRIXG EQUALITY. 1. Quantities which contain the same nnit an Formnias equal number of times, are equal. ^ '''5., 2. Things which being applied to each other Equality. o o 1 1 coincide, are equal in all their parts. CHAP. I.] COMPAEISGN OF QUANTITIES. 117 3. Things wliicli are equal to tlie same thing are equal to one another. 4. A whole is equal to the sum of all its parts 5. If eqnals be added to equals, the sums are equal. 6. If equals be taken from equals, the remain- ders are equal. AXI03IS FOR IXFEimiXG I]SrEQUALITY. 1. A whole is greater than any of its parts. 3. If eqnals be added to uneqnals, the sums Formulas for are unequal. inequality. 3. If eqnals be taken from uneqnals, the re- mainders are unequal. § 110. We have thns completed a very brief what fea- tures have and general analytical view of Mathema- been tical Science. We have named and defined the subjects of which it treats — and the forms of the language employed. We have pointed out the character of the definitions, and the na- ture of the elementary and intnitive proposi- tions on which the science rests ; also, the kind of reasoning employed in its creation, and its divisions resulting from tlie use of different symbols and differences of language. We shall now proceed to treat the branches separately. sketched. S o C3 o ^ 1— 1 CO r-< iW Sd 1-^ o 1— ( (=1 ^"o w B o e^ -^ o <^ 'UIAP. IX.] ARITHMETIC FIRST NOTIONS. 119 CHAPTER II. ARITHMETIC SCIENCE AND ART OF NUMBERS. SECTION I. INTEGKAL UNITS FIRST NOTIONS OF NUMBERS. § 111. There is but a single elementary idea But one eio , . r 1 • • 1 • 1 r y mentary idea in the science oi numbers: it is the idea of the lu numbers. UNIT ONE. There is but one way of impressing Howim- this idea on the mind. It is by presenting to ^thTmm^d! the senses a single object ; as, one apple, one peach, one pear, &;c. .5 112. There are three signs by means of Threesigns . , for express- which the idea of one is expressed and commu- ingu. nicated. They are, 1st. The word one. a word. 2d. The Roman character I. Roman o 1 mi n character-, od. The figure I. '= . Figure. 120 MATHEMATICAL SCIENCE [bOOK II New ideas § 113. If oiie be added to one, the idea thus which iii'isG by adding arising is different from the idea of one, and is °^^' complex. This new idea has also three signs ; viz. TWO, II., and 2. If one be again added, that is, added to two, the new idea has likewise three signs ; viz. three, III., and 3. These Collections collections, and similar ones, arc called num- are num- ^ bers. bers. Hence, A NUMBER is a unit or a collection of units. IDEAS OF NUMBERS GENERALIZED. Ideas of § 114. If wc begin with the idea of the num- generaiized. ^cr onc, and then add it to one, making two ; and then add it to two, making three ; and then to thi'ee, making four ; and then to four, making How formed, fivc, and SO On ; it is plain that we shall form a series of numbers, each of which will be greater Hnity the bv onc than that which precedes it. Now, one or unity, is the basis of this series of numbers. Three ways of expressing and each number may be expressed in three them. ways : 1st way. 1st. By the words one, two, three, &c., of our common language ; 2d way. 2d. By the Romaii characters ; and, 3d way. 3d. By figures. CHAP. II.] .ARITHMETIC UNITY. 12,1 notions are complex. § 115. Since all numbers, whether integral or AUanmbers ^ . , P 111 come from rractional, must come irom, and hence be con- ^ne: nected with, the unit one, it follows that there is but one purely elementary idea in the science of numbers. Hence, the idea of every number, Hence but 11 I r • / 1 11 1 one idea thai regarded as made up oi units (and all numbers is purely eie- except one must be so regarded when we ana- '^™'^''-^- lyze them), is necessarily complex. For, since another the number arises from the addition of ones, the apprehension of it is incomplete until we under- stand how those additions were made ; and there- fore, a full idea of the number is necessarily com.plex. § 116. But if we regard a number as an en- tirety, that is, as an entire or whole thing, as an entire two, or three, or four, without pausing to when a , , . f, , . , . . , . number may analyze the units oi which it is made up, it may ^^, regarded then be regarded as a simple or incomplex idea ; asmcompiex. though, as we have seen, such idea may always be traced to that of the unit one, which forms the basis of the number. UNITY AND A UNIT DEFINED. § 117. When we name a number, as twenty what is ne- feet, two things are necessary to its clear appre- ^''^*^'y'°'*^'' ^ "^ '^ apprehension hension. ©I a number 122 MATHEMATICAL SCIENCE. [book II. First. 1st. A distinct apprehension of the single thing which forms the base of the number ; and, Second. 2d. A distinct apprehension of the number oj times which that thing is taken. The basis of The single thing, which forms the base of the the number i • ii i t^ • ii j u UNITY, numbei', is called unity, or a unit, it is called When it is Unity, when it is regarded as the primary base called UNITY, ^j- ^j^g number ; that is, when it is the final stand- ard to which all the numbers that come from it are referred. It is called a unit when it is re- garded as one of the collection of several equal thinsrs which form a number. Thus, in the ex- ample, one foot, regarded as a standard and the base of the number, is called unity; but, con- sidered as one of the twenty equal feet which make up the number, it is called a unit. and when a UNIT. Abstract unit. OF SIMPLE AND DENOMINATE NUMBERS. § 118. A simple or abstract unit, is one, with- out regard to the kind of thing to which the term one may be applied. Denominate ^ denominate or concrete unit, is one thing ""''• named or denominated ; as, one apple, one peach, one pear, one horse, &c. Number has no reference § 119. Number, as such, has no reference to the particular things numbered. But to dis- CHAP. II.] ARITHMETIC ALPHABET. 123 tinguish numbers which are applied to particular to the things units from those which are purely abst.iact, we call the latter Abstract or Simple Numbers, simple and the former Concrete or Denominate Num- Denominate. bers. Thus, fifteen is an abstract or simple number, because the unit is one ; and fifteen Examples. pounds is a concrete or denominate number, because its unit, one pound, is denominated or named. ALPHABET WORDS GRAMMAR. § 120. The term alphabet, in its most general Alphabet sense, denotes a set of characters which form the elements of a written language. When any one of these characters, or any vvonia. combination of them, is used as the sign of a distinct notion or idea, it is called a word ; and the naming of the characters of which the word is composed, is called its spelling. Grammar, as a science, treats of the estab- Grammai lished connection and relation of words, as the signs of ideas. ARITHMETICAL ALPHABET. § 121. The arithmetical alphabet consists of Arithmetical Alphabet ten characters, called figures. 1 hey are. Naught, One, Two, Throe, Four, Five, Six, Seven, Eight, Nine, 012345678 9 124 MATHEMATICAL SCIENCE. [bOOK II. and each may be regarded as a word, since it stands for a distinct idea. WORDS — SPELLING AND READING IN ADDITION. ono cannot § 122. The idea of one, being elementary, the e ape e . ^.j^^j.^^j^gj. j ^yhich represents it, is also element- ary, and hence, cannot be spelled by the other characters of the Arithmetical Alphabet (§ 121). But the idea which is expressed by 2 comes from Spelling by the addition of 1 and 1 : hence, the word repre- the nritbraeticai sented by the character 2, may be spelled by Characters, j ^^^^ j_ rpj^^^^ j ^^^^ j ^^,g 2, is the arithmet- ical spelling of the word two. Three is spelled thus : 1 and 2 are 3 ; and also, 2 and 1 are 3. Eiiampies. YouY is spelled, 1 and 3 are 4 ; 3 and 1 are 4 ; 2 and 2 are 4. Five is spelled, 1 and 4 are 5 ; 4 and 1 are 5 ; 2 and 3 are 5 ; 3 and 2 are 5. Six is spelled, 1 and 5 are 6 ; 5 and 1 are 6 ; 2 and 4 are 6 ; 4 and 2 are 6 ; 3 and 3 are 6. All numbers § 123. In a similar manner, any number in "^ledina ai'lthmctic may be spelled; and hence we see Bimiiarway. ^j-,^t ^j-jg process of spelling in addition consists simply, in naming any two elements which will make up the number. All the numbers in ad- CHAP 11. J ARITHMETIC READINGS. 125 ditiun are therefore spelled with two syllables. The reading consists in naming only the word Reading: in which expresses the final idea. Thus, gists. 0123456789 Examplea. 1111111111 One two three four five six seven eiglit nine ten. We may now read the words which express. the first hundred combinations. READINGS. Read. 1 2 3 4 5 6 7 8 9 10 Two, three, 1 1 1 1 1 1 1 1 1 1 foui-, &c. 1 2 3 4 5 6 7 8 9 10 Three, four, 2 2 2 2 ■ 2 2 2 2 2 2 &c. 123456789 10 Four, five, 3333333333 &c. 12345678 910 Five,six,&Q 4444444444 123456789 10 six, seven, 55555555 5 5 &c. 123456789 10 seven, eight, 6 6 6 6 6 6 6 6 6 6 *'''• 123456789 10 Eight,nine, 7777777777 ^''• 123456789 10 Ninc,ten,&<% 8888888888 126 MATHEMATICAL SCIENCE. [book II Ten, eleyen, 1 &c. r» Eleven, twelve, &c. 8 9 10 9 9 9 1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 10 Example for § 124. In this example, beginning reading in • i i Addition, at the right hand, we say, 8, 17, 18, 26 : setting down the 6 and carry- ing the 2, we say, 8, 13, 20, 22, 29 : setting down the 9 and carrying the 2, we say, 9, 12, 18, 22, 30: and setting down the 30, we have the entire sum All examples 3096. All the examples in addition may be done BO solved. in a similar manner. 878 421 679 354 764 3096 Advantages of reading. § 125. The advantages of this method of read- ing over spelling are very great, lat. stated. 1st. The mind acquires ideas more readily through the eye than through either of the other senses. Hence, if the mind be taught to appre- hend the result of a combination, by merely see- ing its elements, the process of arriving at it is much shorter than when those elements are pre- sented through the instrumentality of sound. Thus, to see 4 and 4, and think 8, is a very dif- ferent thing from saying, four and four are eight. 2d. The mind operates with greater rapidity and certainty, the nearer it is brought to the Sd. stated. CHAP, ir.] ARITHSIETIC WORDS. 127 ideas which it is to apprehend and combine. Therefore, all unnecessary words load it and impede its operations. Hence, to spell when we can read, is to fill the mind with words and sounds, instead of ideas. 3d. All the operations of arithmetic, beyond 3d. staled the elementary combinations, are performed on paper ; and if rapidly and accurately done, must be done through the eye and by reading. Hence the great importance of beginning early with a method which must be acquired before any con- siderable skill can be attained in the use of figures. § 12G. It must not be supposed that the read- Reading 1 Til -11 77-1 comes attci ing can be accomplished until the spelling has spelling. first been learned. In our common language, we first learn the same asm . our common alphabet, then we pronounce each letter m a language word, and finally, we pronounce the word. We should do the same in the arithmetical reading. WORDS SPELLING AND READING IN SUBTRACTION. § 127- The processes of spelling and reading samo piinci- which we have explained in the addition of insuwrao numbers, may, with slight modifications, be ap- '^'"^ plied in subtraction. Thus, if we are to subtract 128 MATHEMATICAL SCIENCE, [boc 2 from 5, we say, ordinarily, 2 from 5 leaves 3 ; or 2 from 5 three remains. Now, the word, three, is suggested by the relation in which 2 and 5 stand to each other, and this word may be Readings in read at oncc. Hence, the reading, in suhtrac- Subtraction . . . , . , 7 > • 7 explained, tion, IS Simply naming the word, which expresses the difference between the subtrahend and min- uend. Thus, we may read each word of the folIowinfT one hundred combinations. READINGS One from 1 2 3 4 5 G 7 8 9 10 one, &c. 1 1 1 1 1 1 1 1 1 1 Two from 2 3 4 5 6 7 8 9 10 11 two, &c. 2 2 2 2 2 2 2 2 Three from 3 4 5 G 78 9 10 11 12 three, &c. 3333333333 Fourfrom 4 5 G 7 8 9 10 11 12 13 four, &c. 4444444444 Fivefrom 5 G 7 8 9 10 11 12 13 14 fiye,&c. 5555555555 Six from six, G 7 8 9 10 1 1 12 13 14 15 ^'^ GGGGG6GGG6 Seven from 7 8 9 10 11 12 13 14 15 16 seven, &c. 7777777777 CHAP. II.] ARITHMETIC SPELLING. 129 8 9 10 11 12 13 14 15 16 17 Eight from 8888888888 ''^^^^' ^"^ 9 10 11 12 13 14 15 16 17 18 Nine from 9999999999 niue,&c. 10 11 12 13 14 15 16 17 18 19 Ten from ten, 10 10 10 10 10 10 10 10 10 10 ^'^ § 128. It should be remarked, that in subtrac- tion, as well as in addition, the spelling of the speinng pre- ., 1 1 • !• cedes reading words must necessarily precede their reading, insubtrao- The spelling consists in naming the figures with '°°* which the operation is performed, the steps of the operation, and the final result. The reading Reading, consists in naming the final result only. SPELLING AND READING IN MULTIPLICATION. § 129. Spelling in multiplication is similar to Spelling in ■ ,. . , ,. . , Multiplica- tne corresponding process in addition or subtrac- uou. tion. It is simply naming the two elements which produce the product ; whilst the reading Reading. consists in naming only the word which ex- presses the final result. In multiplying each number from 1 to 10 by Examples in 2, we usually say, two times 1 are 2 ; two times ®p""* 2 are 4 : two times 3 are 6 ; two times 4 are 8 ; two times 5 are 10; two times 6 are 12; two 9 130 MATHEMATICAL SCIENCE. [bOOK II, times 7 are 14; two times 8 are IG; two times In reading. 9 are 18; two times 10 are 20. Whereas, we should merely read, and say, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. In a similar manner we read the entire mul- tiplication table. READINGS. Onceoneia 12 11 10 987654321 1, &c 2 Twotiraesi 12 1 1 10 9 8 7 6 5 4 3 2 1 are 2, &c. O Threetimesl 12 1 1 10 9 8 7 6 5 4 3 2 1 are 3, &c. o Fourtimesl 12 11 10 9 8 7 6 5 4 3 2 1 are 4, &.c. ^ Fivetimcsi 12 11 10 9 8 7 6 5 4 3 2 1 are 5, &c. g sixtimesi 12 11 10 9 8 7 6 5 4 3 2 1 are six, &.C. Q Seventimes 12 1 1 10 9 8 7 6 5 4 3 2 1 I are 7, &c. 7 Eight times 1 are 8, &.c. 12 11 10 9 8 7 6 5 4 3 2 1 CHAr. II.] ARITHMETIC READING. 131 12 11 10 9 8 7 G 5 4 3 2 1 Nine times l q are 9, &e. 12 11 10 9 8 7 G 5 4 3 2 1 Xentimesl 1 n are 10, &c 12 11 10 9 8 7 G 5 4 3 2 1 Eleventimes 1 I 1 are 11, &e. 12 11 10 9 8 7 G 5 4 3 2 1 Xwelvetimw 22 lnrel2,&c. SPELLING AND READING IN DIVISION. § 130. In all the cases of short division, the inshortoivi- , , . 1 • 1 • I sioii, we may quotient may be read immediately without nam- read: ing the process by which it is obtained. Thus, in dividing the following numbers by 2, we merely read the words below. 2)4 G 8 10 12 IG 18 22 two three four five six eight uiue eleven. In a similar manner, all the words, expressing in aii cases, the results in short division, may be read. READINGS. 2)2 4 G , 8 10 12 14 16 18 20 22 24 two ma, once, &c 3)3 6 9 12 15 18 21 24 27 30 33 36 xhr^eina, once, &c. 4)4 8 12 16 20 24 28 32 36 40 44 48 Four in 4, once, &C. 132 MATHEMATICAL SCIENCE. [bOOK 11. FiTeins, 5)5 10 15 20 25 30 35 40 45 50 55 60 once, &c. Six in 6, 6)6 12 18 24 30 36 42 48 54 60 66 72 once, &c. Seven in 7, 7)7 14 21 28 35 42 49 56 63 70 77 84 once, &c. Eight in 8, 8)8 16 24 32 40 48 56 64 72 80 88 96 once, &.C. Nine in 9, 9) 9 18 27 36 45 54 63 72 81 90 99 108 once, &c. Ten in 10, 10)10 20 30 40 50 60 70 80 90 100 110 120 once, &c. Elleven in 11, 11)11 22 33 44 55 66 77 88 99 110 121 132 once, &c. Twelve in 12, 12)12 24 36 48 60 72 84 96 108 120 132 144 once, &.C. UNITS INCREASING BY THE SCALE OF TENS. The idea of a § 131. The idea of a particular number is ne- number is cessarilv complex ; for, the mind naturally asks : compex. jg^^ What is the unit or basis of the number? and, 2d. How many times is the unit or basis taken ? What a fig- § 1-32. A figure indicates how many times a Qie indicates. . . , _, , . , _ , unit IS taken, iiiacn or the ten ngures, however written, or however placed, always expresses as many units as its name imports, and no more ; nor does the Jigure itself at all indicate the kind CHAP. II.] ARITHMETIC SCALE OF TENS. lo3 cf unit. Still, every number expressed by one or Number has one for ita more figures, has for its base either the abstract basis, unit one, or a denominate unit.* If a denomi- nate unit, its value or kind is pointed out either by our common language, or as we shall present- ly see, by the j)lo-ce where the figure is written. The number of units which may be expressed by either of the ten figures, is indicated by the Number ex- name of the figure. If the figure stands alone, single figure. and the unit is not denominated, the basis of the number is the abstract unit 1. 8 133. If we write on the rijrht of j ^ "" [ 10, How ten is 1, we have ) written. which is read one ten. Here 1 still expresses ONE, but it is one ten ; that is, a unit ten times as great as the unit 1 ; and this is called a unit unit of the „ , J , second order. oi the second order. Acrain ; if we write two O's on the , „ * -. ^ ' f T /-v/% How to wnte , 100, right of 1, we have ^ one hundred. which is read one hundred. Here again, 1 still expresses one, but it is one hundred; that is, a unit ten times as great as the unit one ten, and a unit of the , 11- ,1 • . 1 third order. a hundred tuT^es as c-reat as tlie unit 1. § 134. If three I's are written by Laws — when IT] figures are the side of each other, thus . - . -S ' ^""en by the side of each other, * Section 118. 134 MATHEMATICAL SCIENCE. [book II the ideas, expressed in our common anguage, are these : First. 1st. That the 1 on the right, will either express a single thing denominated, or the abstract unit one. Second. 2d. That the 1 next to the left expresses 1 ten that is, a unit ten times as great as the first. Thiid. 3d. That the 1 still further to the left expresses 1 hundred ; that is, a unit ten times as great as the second, and one hundred times as great as the first ; and similarly if there were other places. What the When figures are thus written by the side of eat'ablis^es cach Other, the arithmetical language estabUshes w'aen figures ^ relation betwccn the units of their places : that are so writ- ^ len. is, the unit of each place, as we pass from the right hand towards the left, increases according to the scale of tens. Therefore, by a law of the arithmetical language, the place of a figure fixes its unit. Scale for If, then, we write a row of I's as a scale, Numeration. , . thus : 13 § g c 3 ,c -2 ;s tH e S a; s ;i o ^ 111, 111, 111, 111 The units of place deter- mined, the unit of each place is determined, as well CHAP. II.] ARITHMETIC SCALE OF TENS. 135 as the law of change in passing from one place to another. If then, it were required to express how any number of a given number of units, of any order, we first units may be select from the arithmetical alphabet the char- ^^"^'^ acter which designates the number, and then write it in the place corresponding to the order. Thus, to express three millions, we write 3000000 ; and similarly for all numbers. § 135. It should be observed, that a figure a figure has no value in being a character wiiicii represents value, can itsey. have no value in and of itself The number of things, which any figure expresses, is determined by its name, as given in the arithmetical alpha- bet. The kind of thing, or unit of the figure, is How the una fixed either by naming it, as in the case of a de- mined, nominate number, or by the place which the figure occupies, when written by the side of or over other figures. The phrase "local value of a figure," so Figure, has no local long in use, is, therefore, without siguificatiou value. when applied to a figure: the term "local value," being applicable to the unit of tlie Term ap- plicable to place, and not to the figure which occupies ihQ unit, of picux. place. Federal § 13G. United States Currency affords an ex- Money: 136 MATHEilATICAL SCIENCE. [BOOK II. Its denom- ample of a series of denominate imits, increasing iuationa. according to the scale of tens : thus, -r !-■" P:J ft P O S 11111 How read, may be read 11 thousand 1 hundred and 11 mills; or, 1111 cents and 1 mill; or, 111 dimes ] cent and 1 mill; or, 11 dollars 1 dime 1 cent and 1 mill ; or, 1 eagle 1 dollar 1 dime 1 cent Various kinds and 1 mill. Thus, we may read the number of Reading. . , . , „ . . , . With either oi its units as a basis, or we may name them all : thus, 1 eagle, 1 dollar, 1 dime, 1 cent, 1 mill. Generally, in Federal Money, we read in the denominations of dollars, cents, and mills; and should say, 11 dollars 11 cents and 1 mill. Examples in § 137. Examples in reading figures : — Reading. \ r istExampie. li wc havc the hgurcs - - - - 89 we may read them by their smallest unit, and say eighty-nine ; or, we may say 8 tens and 9 units. ?d. Example. Again, the figures 567 may be read by the smallest unit ; viz. five hundred and sixty-seven ; or we may say, 56 tens and 7 units ; or, 5 hundreds 6 tens and 7 units. 3d. Example. Again, the number expressed by - 74896 CHAP. II.] ARITHMETIC VARYING SCALES, 137 may be read, seventy-four thousand eight hun- various read- . ■ r^ • '"S3 of a dred and ninety-six. Or, it may be read, 7489 number. tens and 6 units ; or, 748 hundreds 9 tens and 6 units ; or, 74 thousands 8 hundreds 9 tens and 6 units ; or, 7 ten thousands 4 thousands 8 hundreds 9 tens and 6 units ; and we may read in a similar way all other numbers. Although we should teach all the correct read- The best ings of a number, we should not fail to remark reading. that it is generally most convenient in practice to read by the lowest unit of a number. Thus, in the numeration table, we read each period by Each period the lowest unit of that period. For example, in lo'^vestunit. the number 874,967,847.047, Example. we read 874 billions 967 millions 847 thousands and 47. UMTS INCREASING ACCORDING TO VARYING SCALES. § 138. If we write the well-known signs of Methods of the English money, and place 1 under each de- ures having nomination, we shall have _, . , denommate £. S. d. f. 1111 different nomini units. Now, the signs £. s. d. and^^. fix the value of How the the unit I in each denomination; and they also unit is fixed. 138 MATHEMATICAL SCIENCE. [bOOK II. What the determine the relations which subsist between expresses. t^^G different units. For example, this simple ianjTuage expresses these ideas : The units of Ist. That the unit of the right-hand place is e paces. ^ farthing — of the place next to the left, 1 penny — of the next place, 1 shilling — of the next place, 1 pound ; and How the 2d. That 4 units of the lowest denomination increase, make One unit of the next higher; 12 of the second, one of the third ; and 20 of the third, one of the fourth. The units in If w6 take the denominate numbers of the Avoii'dupois weight, we have Avoirdupois weight. Ton. cwt. qr. lb. oz. dr. 111111; Changes in in which the units increase in the following evaueo jj^j^j^j^g^, . yj^. the second unit, counting from the units. ' o the right, is sixteen times as great as the first; the third, sixteen times as great as the second ; the fourth, twenty-five times as great as the third ; the fifth, four times as great as the fourth ; and the sixth, twenty times as great as the fifth. HowthescaJe The scale, therefore, for this class of denominate numbers varies according to the above laws. A different If wc take any other class of denominate eystem! numbers, as the Troy weight, or any of the systems of measures, we shall have different scales for the formation of the different units. CHAP. II.] ARITHMETIC — IITTEGEAL UKITS. 139 But in all the formations, we shall recognise The method , f • r 1 1 • • 1 °^ forming the application oi the same general principles. ^^e scales tht There are, therefore, two general methods of ^™'''°'"^ o munbers. forming the different systems of integral num- Two systems bers from the unit one. The first consists in of forming preserving a constant law of relation between *° "^^y^rT*^™" the different unities ; viz. that their values shall First system. change according to the scale of tens. This gives the system of common numbers. The second method consists in the application second sys- of known, though varying laws of change in the unities. These changes in the unities produce changeintha the entire system of denominate numbers, each formi„gtho class of which has its appropriate scale, and the ""'"ex- changes among the units of the same class are indicated by the different steps of its scale. , INTEGEAL UNITS OF AEITHMETIC. §139. There are eight classes of units — four Eight ciass- p , T r. n ■ 68 of units. 01 number, and tour of space, viz. 1. Abstract Units; 5. Units of Lines; Abstract, 2. Units of Currency; 6. Units of Surface; currency, 3. Units of Weight; 7. Units of Volumes; Weight, 4. Units of Time; 8. Units of Angles. Time. First among the Units of Arithmetic stands the simple or abstract unit 1. This is the base Abstract unit one, the of all abstract numbers, and becomes the base, base. 140 MATHEMATICAL SCIENCE. [bO( The basis of also, of all denominate numbers, by merely na- denominate . . . . , , . numbers; miHg, HI succession, the particular thmgs to which it is applied. Also, the ba- It is also the basis of all fractions. Merely as sis of all frac- tions, the unit 1, it is a whole which may be divided whether sim- , . , ^ . . ^ pieordenom- accoi'ding to any law, lorming every variety oi "^^'®" fraction ; and if we apply it to a particular thing, the fraction becomes denominate, and we have expressions for all conceivable parts of that thing. § 140. It has been remarked* that we can Mustappre- form no distiuct apprehension of a number, un- heud the unit. til we have a clear notion of its unit, and the number of times the unit is taken. The unit is the great basis. The utmost care, therefore, Let its nature should be taken to impress on the minds of and kind be .... fully explain- learners, a clear and distinct idea of the actual ed ' ' value of the unit of every number with which they have to do. If it be a number expressing ow or a Qyj-j.Qj^cy Qj^g Qj- more of the coins should be Dtimber ex- •' ' pressingciir- exhibited, and the value dwelt upon; after which rency. distinct notions of the other units of currency can be acquired by comparison. If the number be one of weight, some unit Exhibit the ^ anitif itbe should be exhibited, as one pound, or one ounce, of weight; and an idea of its weight acquired by actually * Section 110. CHAP. II.] UNITED STATES CIJEEEISrCT. 141 lifting it. This is the only way in which we can learn tlie true signification of the terms. If the number be one of measure, either ^"^^^®°'^^ it be one of linear, superficial, of volumes or of angles, its measm-e. unit sliould also be exhibited, and tlie significa- tion of tlie term expressing it, learned in the only way in wldcli it can he learned, througli the senses, and by tlie aid of a sensible object. UXITED STATES CUERENCT. § 141. The currency of the United States is currency of called United States Currency. Its units are all states. denominate, being 1 mill, 1 cent, 1 dime, 1 dollar, 1 eagle. The law of change, in passing from one Law of , . , . , 1 r change in ths unit to another, is according to the scale or tens, unities. Hence, this system of numbers may be treated, •^ J ' How these in all respects, as simple numbers; and indeed '^'''"ibera may be treated. they are such, with the single exception that their units have different names. They are generally read in the units of dollars. How gen- cents, and mills — a period being placed after *^'^ ^^^^ ' the figure denoting dollars. Thus, ^864.849 Example. is read eight hundred and sixty-four dollars, eighty-four cents, and nine mills ; and if there were a figure after the 9, it would be read in °''''°"''^' ^ after milla decimals of the mill. The number may, how- 142 MATHEMATICAL SCIENCE. [bOOK II. The number evci', be Tcad in any other unit ; as, 864849 read in vai-ious ways, mills ; OT, 86484 cents and 9 mills; or, 8648 dimes, 4 cents, and 9 mills ; or, 86 eagles, 4 dol- lars, 84 cents, and 9 mills ; and there are yet several other readings. ENGLISH MONEY. Sterling Mo- § 142. The units of English, or Sterling Mo- ^^' ney, are 1 farthing, 1 penny, 1 shilling, and 1 pound. scaieofthe The scale of this class of numbers is a varying scale. Its steps, in passing from the unit of the lowest denomination to the highest, are four. How it twelve, and twenty. For, four fsrthings make one penny, twelve pence one shilling, and twenty shillings one pound. unities. changes. AVOIRDUPOIS WEIGHT. Units in § 143. The units of the Avoirdupois Weight Avoirdupois. , i , , i , , are 1 dram, 1 ounce, 1 pound, 1 quarter, 1 hun- dred-weight, and 1 ton. Scale. The scale of this class of numbers is a vary- ing scale. Its steps, in passing from the unit of the lowest denomination to the highest, are sixteen, sixteen, twenty-five, four, and twenty. Variation in ■^^^' sixteen drams make one ounce, sixteen ita degrees, ounccs One pound, twenty-fivc pounds one quar- CHAP. II.] ARITHMETIC — UKITS OF LEISTGTH. 143 ter, four quarters one liundred, and twenty hun- dreds one ton. TROT WEIGHT. § 144. The units of the Troy "Weight are, 1 units in Troy grain, 1 pennyweight, 1 ounce, and 1 pound. Weight. The scale is a varying scale, and its steps, in Scale: passing from the unit of the lowest denomina- its degrees. tion to the highest, are twenty-four, twent}', and twelve. apothecaries' WEIGHT. § 145. The units of this weight are, 1 grain, 1 Units in Apotheca- scruple, 1 dram, 1 ounce, and 1 pound. ries' Weight. The scale is a varying scale. Its steps, in scaie: passing from the unit of the lowest denomina- its degrees, tion to the highest, are twenty, three,, eight, and twelve. units of measure of space. § 146. There are four units of measure of ronr units of measure. Space, each differing in Icincl from the other three. They are. Units of Length, Units of Surface, Units of Volume, and Units of Angular Measure. units of length. § 147. The unit of length is used for measur- Units of ing lines, either straight or curved. It is a kugth. 144 MATHEMATICAL SCIENCE. [bOOK If. The stand- straight line of a given length, and is often called bli- the standard of the measuren:ient. The units of length, generally used as stand- ards, are 1 inch, 1 foot, 1 yard, 1 rod, 1 furlong, and 1 mile. The number of times which the unit, used as a standard, is taken, considered in connection with its value, gives the idea of the length of the line measured. What units ure taken. Idea of length. UNITS OF SUKFACE. Units of surface. What the unit of surface is. Examples. Ita connection with the unit of length. Square feet in a square yard. 1 square foot. § 148. Units of surface are used for the meas- urement of the area or contents of whatever has the two dimensions of length and breadth. The unit of surface is a square de- scribed on the unit of length as a side. Thus, if the unit of length be 1 foot, the corre- sponding unit of surface will be 1 square foot ; that is, a square constructed on 1 foot of length as a side. ' If the linear unit be 1 yard, the corresponding unit of sur- face will be 1 square yard. It will be seen from the figure, that, although the linear yard contains the linear foot but three times, the square yard 1 yard. CHAP. II. J ARITHMETIC DUODECIMAL UNITS. 145 contains the square foot nine times. The square Square rod rod or square mile may also be used as the unit square miie. of surface. The number of times which a surface contains Area or P . . , contents of a its unit oi measure, is its area or contents ; and gurface. this number, taken in connection with the value of the unit, gives the idea of its extent. Besides the units already considered, there is a special class, called DUODECIMAL UNITS. § 149. The duodecimal units are generally used Duodecimal in board and timber measure, though they may be used in all measurements of surface and volume. They are simply the units 1 foot, 1 square foot, what they and 1 cubic foot, divided according to the scale of 12. § 150. It is proved in Geometry, that if the -^v^jat pnnci- number of linear units in the base of a rectan- ju'ceometry. gle be multiplied by the number of linear units in the breadth, the numerical value of the pro- duct will be equal to the number of superficial units in the figure. Knowing this fact, we often express it by say- How it is ex. ing, that "feet multiplied by feet give square ^'^^^^ ' feet," and "yards multiplied by yards give square 10 146 MATHEMATICAL SCIENCE. [book 11. ThUaconcise yards." But as feet cannot be taken jTee^ timss, *"'^' .' ' nor yards, yard times, this language, rightly un- derstood, is but a concise form of expression for the principle stated above. Conclusion. With tliis Understanding of the language, we say, that 1 foot in length multiplied by 1 foot in breadth, gives a square foot; and 4 feet in length multiplied by 3 feet in breadth, gives 12 square feet. Kxumples in the mullipli- cation of feet by feet and inches. Generaliza- tion. Inches by inches. How the units change, and wh-at they are. First. Second. § 151. If now, 1 foot in length be multiplied by 1 inch =j2 of ^ foot in breadth, the product will be one-twelfth of a square foot; that is, one- tiuelfth, oftlie seccmd unit : if it be multiplied by 3 inches, the product will be three-twelfths of a square foot ; and similarly for a multiplier of any number of inches. If, now, we multiply 1 inch by 1 inch, the product may be represented by I square inch : that is, iy one-hvelfth of one-twelfth of a square foot. Hence, tJie units of this measure decrease accordi7ig to the scale of 12. The units are, 1st. Square feet — arising from multiplying feet by feet. 2d. Twelfths of square feet — arising from mul- tiplying feet by inches. CHAP. II.] AEITHMETIC— UNITS, 147 3d. Twelfths of twelfths — arising from multi- Third, plying inches by inches. When we introduce the third dimension, height, we have, 1 foot being the nnit, 1x1x1 = 1 . Conclusion cubic foot ; 1 X 1 X tV = T^J cubic foot ; 1 X iV X general. T 2 = T44 cubic foot ; and Jg- X yV X ^V = ttW cubic foot. Hence, the units change by the scale of 13. UNITS OF VOLUME. § 152. It has already been stated, that if Units or volume. length be multiplied by breadth, the product may be represented by units of surface. It is what is . proved in also proved, m Geometry, that if the length. Geometry in breadth, and height of any regular figure, of a them, square form, be multiplied together, the pro- duct may be represented by units of volume Units or whose number is equal to this product. Each unit is a cube constructed on tlie linear unit as an edge. Thus, if the linear unit be 1 foot, the Examples, unit of volume will be 1 cubic foot; that is, a cube constructed on 1 foot as an edge; and if it be 1 yard, the unit will be 1 cubic yard. The three units, viz. the unit of length, the The three unit of surface, and the unit of volume, are es- tiaiiy differ- sentially different in kind. The first is a line ^" ' of a known length ; the second, a square of a what they known side ; and the third, a fio^ure, called a 148 MATHEMATICAL SCIENCE. [uOOK IT. Generally cube, of a known base and height. These are the units used in all kinds of measurement — Duixiccimai excepting only angles, and the duodecimal sys- system. . tem, Avhich has already been explained. LIQUID BIEASURE. Units of Li- § 15.3. The units of Liquid Measure are, 1 quid Meas- . n i i i ure. gill, 1 pint, 1 quart, 1 gallon, 1 barrel, 1 hogs- head, 1 pipe, 1 tun. The scale is a varying scale. Its steps, in passing from the unit of Howitva- the lowest denomination, are, four, two, four, thirty-one and a half, sixty-three, two, and two. Scale. nes. DRY MEASURE. Units ftf Dry § 154. The units of this measure are, 1 pint, Measure. 1 quart, 1 peck, 1 bushel, and I chaldron. The Degrees of steps of the scale, in passing from units of the lowest denomination, are two, eight, four, and thirty-six. Units of § 155. The units of Time are, 1 second, 1 Time. -11 minute, 1 hour, 1 day, 1 week, 1 month, I year, iDegreesof and 1 century. The steps of the scale, in the scale. • n • r i i i • ■ passing irom units oi the lowest denomination to the highest, are sixty, sixty, twenty-four, .seven, four, twelve, and one hundred. CHAP. II ] ARITHMETIC ADVANTAGES. 149 ANGULAK, OE CIRCULAR MEASURE. 8 156. The units of this measure are, 1 sec- units of cir- cular Meas- ond, 1 minute, 1 degree, 1 sign, 1 circle. The ure. steps of the scale, in passing from units of the Degrees oi ^ . . . the Scale, lowest denomination to those of the higher, are sixty, sixty, thirty, and twelve. ADVANTAGES OF THE SYSTEM OF UNITIES. § 157. It may well be asked, if the method Acivimtases of thesjsieiB here adopted, of presenting the elementary prin- ciples of arithmetic, has any advantages over those now in general use. It is supposed to pos- sess the following : 1st. The system of unities teaches an exact ist. Teaches analysis of all numbers, and unfolds to the mind of numbers: the different ways in which they are formed from the unit one, as a basis. 2d. Such an analysis enables the mind to form sd. Pointsout a definite and distinct idea of every number, by relation: i pomting out the relation between it and the unit from which it was derived. 3d. By presenting constantly to the mind the 3d. Constant ly pi'piients idea of the unit one, as the basis of all numbers, the idea of the mind is insensibly led to compare this unit *™'^' with all the numbers which flow from it, and 150 MATHEMATICAL SCIENCE. [UOOK II. then it can the more easily compare those num- bers with each other. 4th. Ex- 4th. It affords a more satisfactory analysis, ^fuiiyTiie ^iicl a better nndcrstanding of the four ground ^°"mkr"*^ rules, and indeed of all the operations of arith- metic, than any other method of presenting the subject. Primary bape of By B tern. Scale. METEIC, OK FEEXCH SYSTEM OF WEIGHTS AND MEASURES. § 158. The primary base, in this system, for all denominations of weights and measures, is the one-ten-millionth part of the distance from the equator to the pole, measured on the earth's surface. It is called a Metee, and is equal to 39.37 inches, yery nearly. The change from the base, in all the denom- inations, is according to the decimal scale of tens: that is, the units increase ten times, at each step, in the ascending scale, and decrease ten times, at each step, in the descending scale. MEASURES OF LENGTH. Base, 1 metre = 39.37 inches, nearly. CHAP. II.j METEIC SYSTEM. 151 TABLE Ai Bcend^ ing Scale. Descending Scale. A A / ■* ^ N H ^ B g3 CD o B o H a3 CD B >-> o c3 a CU ^ ^ s w G ^ p o s 1 1 1 1 1 1 1 1 The names, in the ascending scale, are formed Names ia the scale. by prefixing to tlie base, Metre, the words, Deca (ten), Hecto (one hundred). Kilo (one thousand), Myria (ten thousand), from the Greek numerals; and in the descending scale, by prefixing Deci (tenth), Ceuti (hundredth). Mill (thousandth), from the Latin numerals. SQUARE MEASURE. Base, 1 Are = Ihe square whose side is 10 metres. = 119.G square yards, nearly. — 4 perches, or square rods, nearly. The unit of surface is a square whose side is 10 metres. It is called an Are, and is equal to 100 square metres. MEASURE OF VOLUMES. Base, 1 Litre = the cube on tlie decimetre. = G1.023378 cubic inches. = a little more than a wine quart. 152 MATHEMATICAL SCIENCE. [BOOK 11, The unit for the measure of vohiine is the cube whose edge is one-tenth of the metre — that is, a cube whose edge is 3.937 inches. This cube is called a Litke, and is one- thousandth part of the cube constructed on the metre, as an edo-e. FOUK GEOUND RULES. System g 159. Let us take the two following examples applied in addition, in Addition, the one in simple and the other in denominate numbers, and then analyze the pro- cess of finding the sum in each. Examples. SEffPLE NUMEEKS. DENOMINATE NUMBBKS. 874198 cwt. qr. lb. . oz. dr. 36984 3 3 24 15 14 3641 6 3 23 14 8 914823 10 3 23 14 Processor In both examples we begin by adding the units ''addiUoi"" of the lowest denomination, and then, we divide their sum hy so many as mahe one of the denomi- nation next higher. We then set down the remainder, and add the quotient to the units of that denomination. Having done this, we apply a similar process to all the other denomina- tions — the 2^rincij)le leing jyrecisely the same in principle. j^ifA exanijjles. We see, in these examples, an CHAP. II.] ARITHMETIC — SUBTRACTION". 153 illustration of a general principle of addition, units of the • /- /7 7-7 7 777 Bume kiiid VIZ. that units of the same kind are always added unite. together. § 160. Let us take two similar examples in system applied in bubtraction. subtraction. SIMPLE NUMBEKS. DENOMINATE NUMBERS. 8103 £ s. d. far. 3298 6 9 7 2 Examples. 5105 3 10 8 4 2 18 10 2 In both examples we begin with the units of The method of perform- the lowest denomination, and as the number in iugthecx- tlie subtrahend is greater than in the place di- rectly above, we suppose so many to be added in the minuend as make one unit of tlie next higher denomination. We then make the sub- traction, and add 1 to the units of the subtrahend next higher, and proceed in a similar manner, through all the denominations. It is plain that the principle employed is the same in both exam- principle the pies. Also, that units of any denomination in examples, the subtrahend are taken from those of the same denomination in the minuend. § 161. Let us now take similar examples in Muitipiica- Multiplication. '^''°- 154 MATHEMATICAL SCIENCE. [BOOK II. SIMPLE NTJMBESS. DENOMINATE NTJMBEES. imples. 87464 5 437320 9 7 9 gr. G 3 15 5 48 3 2 1 15 Method of lu tliese examplcs we see, that we multiply, in performing . , , „ -i-ji ■,,•■,• the exam- successiOD, eacii Order 01 units in the mnltipli- P^***- cand by the mnltiplier, and that we carry from one product to another, one for every so many as make one unit of the next higher denomination. The princi- ple the same The ^principle of the process is therefore the same for all ex- • n , i i ampius. ^11 l^oth examples. Division. Examples. § 163. Finally, let us take two similar exam- ples in Division. SIMPLE NUMBEKS. DENOMINATE NUMBERS. 3 )8749 11 £ s. d. far. "391637 3)8 4 3 1 3 14 8 3 Principles We bcgiu, in both examples, by dividing the units of the highest denomination. The unit of the quotient figure is the same as that of the dividend. We write this figure in its place, and then reduce the remainder to units of the n( xt The same as ■^*^^^^^' denomination. "We then add in that de- in the other jjominatiou, and continue the division through rules. ' ° all the denominations to the last — the principle being precisely the same in both examples. CHAP, 11.] ARITHMETIC FRACTIOVS. 155 SECTION II. FB ACTIONAL UNITS. FRACTIONAL UNITS. SCALE OF TENS. § 1G3. If the unit 1 be divided into ten equal Fraction one. parts, each part is called one tenth. If one of defined- these tenths be divided into ten equal parts, ^ ^ One each part is called one hundredth. If one of the hundredth; hundredths be divided into ten equal parts, each Q„g part is called one thousandth ; and corresponding thousandth. names are given to similar parts, how far soever oeneraiiza- the divisions may be carried. Now, although the tenths which arise from Fractions are whole things. dividing the unit 1, are but equal parts of 1, they are, nevertheless, whole tenths, and in this light may be regarded as units. To avoid confusion, in the use of terms, we Fractional shall call every equal part of 1 a fractional unit. Hence, tenths, hundredths, thousandths, tenths of thousandths, &c., are fractional units, each having a fixed relation to the unit 1, from which it was derived. 156 MATHEMATICAL SCIENCE. [boOK IT. Fractional § 164. Adopting a similar language to that units of the first Older; used in integral numbci's, we call the tenths, frac- der &c. tional units of the fust ordei^ ; the hundredths, fractional units of the second order ; the thou- sandths, fractional units of the third order ; and so on for the subsequent divisions. Lan<'uac'e for ^^ there any arithmetical language by which fractional ^j^ggg fractional units may be expressed ? The units. •' '■ decimal point, which is merely a dot, or period. What it fixes, indicates the division of the unit 1, according to the scale of tens. By the arithmetical language, Names of the the uijit of the place next the point, on the right, places. IS 1 tenth ; that of the second place, 1 hun- dredth ; that of the third, 1 thousandth ; that of the fourth, 1 ten thousandth ; and so on for places still to the right. Scale. The scale for decimals,' therefore, is .111111111, &c.; in which the value of the unit of each place is known as soon as we have learned the significa- tion of the language. If, therefore, we wish to express any of the parts into which the unit 1 may be divided, ac- cording to the scale of tens, we have simply to Any decimal ~ . number may sclcct from the alphabet, the figure that will be expressed by this scale, exprcss thc numbev of parts, and then write it in CIIAr. II.] ARITHMETIC FRACTIONS. 157 the place corresponding to i\\Q order of the unit, where any fitjure is Thus, to express four tenths, three thousandths, Avritten. eight ten-thousandths, and six milhonths, we write .403806 ; H^ample. and similarly, for any decimal which can be named. § 165. It should be observed that while the units of place decrease, according to the scale of tens, from left to right, they increase according The units in- ■i r • 1 , f. rn • • cicuse from to the same scale, from right to left. This is the ngiit to left same law of increase as that which connects the units of place in simple nu?nbers. Hence, simple consequence numbers and decimals beino; formed according to the same law, may be written by the side of each other and treated as a single number, by merely preserving the separating or decimal point. Thus, 8974 and .67046 may be written 8974.67046 ; Example. since ten units, in the place of tenths, make the unit one in the place next to the left. FRACTIONAL tTNITS IN GffNERAL. § 1G6. If the unit 1 be divided into two equal AhaK parts, each part is called a half. If it be divided 158 MATHEMATICAL SCIENCE. [bOOK II. A third, into three equal parts, each part is called a third : if it be divided into four equal parts, each part is called a fourth : if into five equal parts, each part is called a fifth ; and if into any number of equal parts, a name is given corresponding to the number of parts. Now, although these halves, thirds, fourths, fifths, &c., are each but parts of the unit 1, they are, nevertheless, in themselves, whole things. Examples. That is, a half is a whole half; a third, a whole third ; a fourth, a whole fourth ; and the same for any other equal part of 1. In this sense, therefore, they are units, and we call them frac- Haveaieia- tional uuits. Each is an exact part of the unit A fourth. A fifth. Generally. These units are whole things. tion to unity, 1, and has a fixed relation to it. Language for fractions. To express the number of equal parts. § 1G7. Is there any arithmetical language by which these fractional units can be expressed ? The bar, written at the right, is the sign which denotes the division of the ui'it 1 into any number of equal parts. If we wish to express the number of equal parts into which it is divided, as 9, for example, we simply write the 9 under the bar, and then the phrase means, that some thing regarded as a whole, has been divided into 9 equal parts. 9 CHAP. II. J ARITHMETIC FRACTIONS. 159 If, now, we wish to express any number of these fractional units, as 7, for example, we place the 7 above the line, and read, seven ninths. To show how many are la lien. § 168. It was observed,* that two things are necessary to the clear apprehension of an inte- gral number. 1st. A distinct apprehension of the unit which forms the basis of the number ; and, 2dly. A distinct apprehension of the number of times which that unit is taken. Three things are necessary to the distinct ap- prehension of the value of any fraction, either decimal or vulgar. 1st. We must know the unit, or whole thing, from which the fraction was derived ; 2d. We must know into how many equal parts that unit is divided ; and, 3dly. We must know how many such parts are taken in the expression. The unit from which the fraction is derived, is called the unit of the fraction ; and one of the equal parts is called, i\\Q fractional unit. For example, to apprehend the value of the Two things necessary to apprehend a niunber. First Second Three things necessary to apprehend a fiaction. Second. Third. Hnit of tho fraction — ef the expres- sion. * Section 117. 160 MAT II EM vriCAL SCIENCE. [book II. viTiatwe fraction f of a pound avoirdupois, or f Z5. ; we raust know. must know, First. Second. Unit when not named. 1st. What is meant by a pound ; 2d. That it has been divided into seven equal parts ; and, 3d. That three of those parts are taken. In the above fraction, 1 pound is the unit of the fraction ; one-seventh of a pound, tlie frac- tional unit; and 3 denotes that three fractional units are taken. If the unit of a fraction be not named, it is taken to be the abstract unit 1. ADVANTAGES OF FRACTIONAL UNITS. Every equal §109. By Considering cvcrj cqual part of uni- part of one, a unit. ty as a unit in itself, having a certain relation to the unit 1, the mind is led to analyze a frac- tion, and thus to apprehend its precise significa- tion. Under this searching analysis, the mind at ones seizes on the unit of the fraction as the principal base. It then looks at the value of each part. It then inquires how many such parts are taken. Equal units. It having bccn shown that equal integral units whetlipr in- tegral or frac- can alone be added, it is readily seen that the Advantages of the analysis. CHAP. II ] ARITHMETIC ADVANTAGES. 161 / same principle is equally applicable to frac- tionai, can , . , , .... , alone be tional units ; and then the inquiry is made : a^de*!. What is necessary in order to make such units equal ? It is seen at once, that two things are neces- ^wo things necessary for sary : addition. 1st. That they be parts of the same unit ; and, ^'^^• 2d. That they be like parts ; in other words, second. they must be of the same denomination, and have a common denominator. In regard to Decimal Fractions, all that is Decimal Fractions, necessary, is to observe that units of the same value are added to each other, and when the figures expressing them are written down, they should always be placed in the same column. S 170. The great difficulty in the management D'fflc""yin the inanage- of fractions, consists in comparing them with ment of frao tions, each other, instead of constantly comparing them with the unit from Avhich they are derived. By consideriijg them as entire things, having a fixed reliition to the unit which is their base, obviated. they can be compared as readily as integral num- bers; for, the mind is never at a loss when it apprehends the unit, the parts into which it is Eeasons for divided, and the number of parts which are greater eim- plicity in taken. The only reasons why we apprehend and integers. 11 163 MATHEMATICAL SCIENCE. [BOOK II. lianclle integral numbers more readily than frac- tions, are, First. 1st. Because the unit forming the base is always kept in view ; and. Second. 2d. Because, in integral numbers, we have been taught to trace constantly the connection between the itnit and the numbers which come from it ; while in the methods of treating frac- tions, these important considerations have beer, neglected. SECTION III. PROPOKTION AND RATIO, Proportion g 171. PROPORTION cxprcsses the relation which defined. one number bears to another, Vv^ith respect to its being; greater or less. Two ways of Two numbcrs may be compared, the one Avith comparing. the other, in two ways : isi method. Ist. With rcspcct to their difference, called Arithmetical Proportion ; and, sdinothod. 2d. With respect to their quotient, called Geometrical Proportion, CHAP. II. J ARITHMETIC PROPORTION. 163 Thus, if we compare the numbers 1 and 8, Exami.ieof by their difference, we find that the second ex- p,'.„py,.'tiy„; ceeds the first by 7 : hence, their difference 7, is the measure of their arithmetical proportion, Arithmetical and is called, in the old books, their arithmetical Ratia ratio. If we compare the same numbers by their Example of quotient, we find that the second contains the ^eometncaj ' Proportion, first 8 times : hence, 8 is the measure of their geometrical proportion, and is called their geo- '^"°" metrical ratio* S 172. The two numbers which are thus com- ° Terms. pared, are called terms. The first is called the Antecedent antecedent, and the second the consequent. consequent. In comparing numbers with respect to their comparison j.fv. ^, i- • 1 1 ■ by difference dirlerence, the question is, now much is one greater than the other ? Their difference affords the true answer, and is the measure of their pro- portion. In comparing numbers with respect to their comparison , . . , . . by ([uotieHt quotient, the question is, now many times is one greater or less than the other ? Their quotient or ratio, is the true answer, and is the measure * The term ratio, as now generally used, means the quo- tient arising from dividing one number by another. We shall use it onlv in this sense. 164 MATHEMATICAL SCIENCE. [boOKII. Example by of their proportion. Ten, for exam])!©, is 9 difference. , . „ , , greater than 1, ii we compare the numbers one and ten by their difference. But if we compare By quotient, them by their quotient, ten is said to be ten "Ten times." times as great — the language "ten times" having reference to the quotient, which is always taken as the measure of the relative value of two Examples of numbers so compared. Thus, when we say, thisuseoftiie ^j^^^ ^j^^ units of our common system of numbers term. •' increase in a tenfold ratio, we mean that they so increase that each succeeding unit shall contain the preceding one ten times. This is a conven- couveoicnt ie^t language to express a particular relation of language. ^^^ numbci's, and is perfectly correct, when used in conformity to an accurate definition. In what § 1*^3. All authors agree, that the measure of g/ the geometrical proportion, between two num- bers, is their ratio ; but they are by no means jjj^ijjjt ^jgg^. unanimous, nor does each always agree with ^'^^^ himself, in the manner of determining this ratio. Some determine it, by dividing the first term by Different me- the second ; others, by dividing the second term by the first.* All agree, that the standard, vvhat- ^^Uutilard the jirisor. ever it may be, should be made the divisor. * The Encyclopedia Metropolitana, a work distinguished by ihe excellence of its scientific articles, adopts the lattei method. CHAP, II.] ARITHMETIC RATIO. 165 This leads us to inquire, whether the mind what is the fixes most readily on the first or second number as a standard ; that is, whether its tendency is to regard the second number as arising from the first, or the first as arising from the second. § 174. All our ideas of numbers begin at Oi-iginof numbeis. one.* This is the starting-point. We con- ceive of a number only by measuring it with now we cob one, as a standard. One is primarily in the '^u^^er' mind before we acquire an idea of any other number. Hence, then, the comparison begins where the at one, which is the standard or unit, and all '=°'"p^'^°" begins. Other numbers are measured by it. When, there- fore, we inquire what is the relation of one to any other number, as eight, the idea presented „ ._, ' ' & ' r The idea is, how many times does eight contain the stand- presented. ard ? We measure by this standard, and the ratio is standai'd. Ratio. the result of the measurement. In this view of the case, the standard should be the first number ^'^^^ ^^^^ should be, named, and the ratio, the quotient of the second number divided by the first. Thus, the ratio of 2 to 6 would be expressed by 3, three benig the Example, number of times which contains 2. ♦Section 111. 11 106 MATHEiMATICAL SCIENCE. [bOOK II Other reasons § 175. The rcasoii fof adopting this method for this me- . .,, -n ^ t thodofoom- of comparison will appear still stronger, it we parison. ^^j,^ fractional numbers. Thus, if we seek the relation between one and one-half, the mind im- mediately looks to the part which one-half is of Comparison o^g^ and this is determined by dividing one-half of unity with fractions, by 1 ; that is, by dividing the second by the first : whereas, if we adopt the other method, we divide our standard, and find a quotient 2. Geometrical § 176. It may be proper here to observe, that proportion. , ., , • i • ;> • i while the term " geometrical proportion is used to express the relation of two numbers, com- Ageometri- pared by their ratio, the term, " a geometrical tion defined, proportiou," is applied to four numbers, in which the ratio of the first to the second is the same as that of the third to the fourth. Thus, Example. 2 : 4 :: 6 : 12, is a geometrical proportion, of which the ratio is 2. Further ad- § 177. We will uow State soinc further ad- \nna-es. ^.^^^^^ggg ^yhJcJ^ result from regarding the ratio as the quotient of the second term divided by the first. Questions in Every question in the Rule of Three is a the Kule of , . • i i i Three- geometrical proportion, excepting only, that the CHAP. II. J ARITHMEnC RATIO. 167 last term is wanting. When that term is found, Their nature. the geometrical proportion becomes complete. In all such proportions, the first term is used as the divisor. Further, for every question in the Rule of Three, we have this ■ clear and simple solution : viz. that, the unknown term or an- how solved. swer, is equal to the third term multiplied by the ratio of the first two. This simple rule, for findino; the fourth term, cannot be given, unless Thisnuede- ^ ^ pends on the we define ratio to be the quotient of the second definition oi Ratio. term divided by the first. Convenience, there- fore, as well as general analogy, indicates this as the proper definition of the term ratio. §178. Again, all authors, so far as I have xhisdeflm- . ^ . i-iii',' '"3" of ratio is consulted them, are uniform m then' definition ^^^ ^^ ^^ of the ratio of a geometrical progression : viz. ^ne ^T that it is the quotient which arises from divid- ing the second term by the first, or any other term by the preceding one. For example, in the progression 2 : 4 : 8 : IG : 32 : 64, &c., Example: all concur that the ratio is 2 ; that is, that it is in which the quotient which arises from dividing the sec- agree. ond term by the first : or any other term by the preceding term. But a geometrical progression differs from a geometrical proportion only in 168 MATHEMATICAL SCIENCE. [BOOK 11, The same this : ill the former, the ratio of any two terms should take . i -i • i i piacemevery IS the sume ; while m the latter, the ratio of the foTthey are ^"'^^ ^^^ secoiid is different from that of the sec- uii the same, ^^-^j ^j^^ third. There is, therefore, no essential difference in the two proportions. Why, then, should we say that in the propor- tion 2 : 4 :: G : 12, E-vampIes. the ratio is the quotient of the first term divided by the second ; while in the progression 2 : 4 : 8 : 16 : 32 : 64, &c., the ratio is defined to be the quotient of the sec- ond term divided by the first, or of any term di- vided by the preceding term ? Wherein As far as I havc examined, all the authors who have defined the ratio of two numbers to be the quotient of the first divided by the sec- ond, have departed from that definition in the case of a geometrical progression. They have How used there used the word ratio, to express the quo- '^**'"' tient of the second term divided by the first, and this without any explanation of a change in the definition. other in- Most of them havc also departed from theii definition, in informing us that " numbers in- authoi'S have depart- ed from their defluitions : stances in which the jefinitionof urease from right to left in a tenfold ratio," in CHAP. II.] ARITHMETIC PROPORTION. 169 which the term ratio is used to denote the quo- Ratio is not tient of the second number divided by the first. The definition of ratio is thus departed from, and the idea of it becomes confused. Such consequen- ces. discrepancies cannot but introduce confusion into the minds of learners. The same term should always be used in the same sense, and have but a single signification. Science does what science I T 1 r I • demands, not permit the slightest departure from this rule. I have, therefore, adopted but a single significa- tion of ratio, and have chosen that one to which xhedeflm- . . tion adopted all authors, so lar as 1 know, have given their sanction ; although some, it is true, have also used it in a different sense. § 179. One important remark on the subject importam r • ■ 1 1 T • 1 • Remark. 01 proportion is yet to be made. It is this : Any two numbers which are compared togeth- Numbers compared er, cither hij their difference or quotient, must must be of I ^ , 7 • 7 7 • I • ; '•'^^ same be of the same knid: that is, they must either i^inj^ have tlte same unit, as a base, or be susceptible of reduction to the same unit. For example, we can compare 2 pounds with Examples 1 ^ • ^• rf • ^ i i • relating to G pounds : their difference is 4 pounds, and their Arithmetic!^,' ratio ii. the abstract number 3. We can also i-icai Propor- compare 2 feet with 8 yards : for, although the '^'""' unit 1 foot is different from the unit 1 yard, still 8 yards are equal to 24 feet. Hence, the differ- 170 MATHEMATICAL SCIENCE. [bOOK II. ence of the numbers is 22 feet, and their ratio the abstract number 12. Numbers On the Other hand, we cannot compare 2 dol- with different units cannot lars with 2 yai'ds of cloth, for they are quantities be com paie J. , . oi different kinds, not being susceptible of reduc- tion to a common unit. Abstract Abstract numbers may always be compared, Qumbers may be corapaj-ed. sincc they have a common unit 1. SECTION IV. APPLICATIONS OF THE SCIENCE OF ARITHMETIC". § 180. Arithmetic is both a science and an Arithmetic: art. It is a sciencc in all that relates to the In what a . . „ , science, properties, laws, and proportions oi numbers. The science is a collection of those connected Science de- processcs whicli dcvclop and make known the fined. laws that regulate and govern all the operations performed on numbers. science per- forms. § 181. Arithmetic is an art, in this : the sci- ence lays open the properties and laws of num- bers, and furnishes certain principles from which CHAP. II.] ARITHMETIC APPLICATIONb 171 practical and useful rules are formed, applicable in the mechanic arts and in business transac- tions. The art of Arithmetic consists in the in what the judicious and ski.ful application of the princi- ples of the science ; and the rules contain the directions for such application. § 1S3. In explaining the science of Arithmetic, in explaining great care should be taken that the analysis of ^hltlilcessa every question, and the reasoning by which the ^^' orinciples are proved, be made according to the strictest rules of mathematical loG;ic. Every principle should be laid down and ex- how each plained, not only with reference to its subsequent s'j'oullfi,''^ use and application in arithmetic, but also, with ^'^'^^'^ reference to its connection with the entire mathe- matical science — of which, arithmetic is the ele- mentary branch. § 183. That analysis of questions, therefore, what 1 . • J '^L iV questions aj» where cost is compared with quantity, or quan- ^^^^^ tity with cost, and which leads the mind of the learner to suppose that a ratio exists between quantities that have not a common unit, is, with- out explanation, certainly faulty as a process of science. For example : if two yards of cloth cost 4 dol- Example. lars, what will 6 vards cost at the same rate ? 172 MATHEMATICAL SCIENCE. [boOKU Analysis: Analysts. — Two yards of cloth will cost twice as much as 1 yard : therefore, if two yards of cloth cost 4 dollars, 1 yard will cost 2 dollars. Again : if 1 yard of cloth cost 2 dollars, G yards, being six times as much, will cost six times two dollars, or 12 dollars. Satisfactory Now, this analysis is perfectly satisfactory to to a child. i -i i tt • • i ■ i a child. He perceives a certain relation between 2 yards and 4 dollars, and between 6 yards and 12 dollars : indeed, in his mind, he compares these numbers together, and is perfectly satisfied with the result of the comparison. Advancing in his mathematical course, how- ever, he soon comes to the subject of propor- tions, treated as a science. He there finds, Reason why greatly to his surprise, that he cannot compare it is defective. i • i i ^■ ly together numbers which have different units ; and that his antecedent and consequent must be of the same kind. He thus learns that the whole system of analysis, based on the above method of comparison, is not in accordance with the prin- ciples of science. True What, then, is the true analysis ? It is this : mialysis : i i' i i i • • G yards of cloth being 3 times as great as 2 yards, will cost three times as much : but 2 yards cost 4 dollars ; hence, 6 yards will cost 3 times 4, or 12 dollars. If this last analysis be not More acien- •' *^'^'=- as simple as the first, it is certainly moie strictly CHAP, ir.J ARITHMETIC APPLICATIONS. 173 scientific ; and when once learned, can be ap- its plied through the whole range of mathematical science. § 184. There is yet another view of this ques- Reasons in tion which removes, to a great degree, if not aret analysis entirely, the objections to the first analysis. It is this : The proportion between 1 yard of cloth and its cost, two dollars, cannot, it is true, as the units are now expressed, be measured by a ratio, according to the mathematical definition of a ratio. Still, however, between 1 and 2, regard- ed as abstract numbers, there is the same relation Numbera existing as between the numbers 6 and 12, also ™"*"'«"''^' " garded as ab regarded as abstract. Now, by leaving out of s"'a<=*= view, for a moment, the units of the numbers, and finding 12 as an abstract number, and then The analysis • ^ • . • ^ 1 , then correct. assignmg to it its proper unit, we have a correct analysis, as well as a correct result. § 185. It should be borne in mind, that practi- How the rules of aiith- cal arithmetic, or arithmetic as an art, selects meticare from all the principles of the science, the mate- °'""'' " rials for the construction of its rules and the proofs of its methods. As a mere branch of yy^^ practical knowledge, it cares nothing about the v^a-cucai ^ ^ ' ^ knowledgo forms or methods of investigation — ^^it demands ;und rules, the two opposite oper- ations of aggregation and divisi(ni are brought into direct contrast with each other. It is thus seen, that the laws of change, in the tAvo systems of opei-ation on the unit 1, are the same with very slight modifications. CHAP. II.] AEITHMETIC — RATIO. 181 This system of classification, has, after expe- rience, been found to be the best for instruction. RATI O, R RULE OF THREE. 8 194. Having considered the two subjects of subjects " considered. integral and fractional units, we come next to the comparison of numbers wnth each other. This branch of arithmetic develops all the what this . . r branch de- relative properties of numbers, resultmg Irom yeiops. their inequahty. The method of arrangement, indicated above, whatiheur rangement presents all the operations of arithmetic ni con- does, nection with the unit 1, which certainly forms the basis of the arithmetical science. Besides, this arrangement draws a broad line what u does .... , . farther. between the science ot arithmetic and its ap- plications ; a distinction which it is very im- portant to make. The separation of the prin- Theory and , . T • practice ciples of a science from their applications, so should be that the learner shall clearly perceive what is ^''P'^^''^- theory and what practice, is of the highest im- portance. Teaching things separately, teaching Golden rules , 11 1 • • 1 • X- forteacliing. them well, and pointing out their connections, are the golden rules of all successful instruc- tion. 195. I had supposed, that the place of the 182 MATHEMATICAL SCIENCE. [nOOK II. Rule of Three, among the branches of arith- metic, had been fixed long since. But several Differences in authors of late, havc placed most of the practi- arrangement; i , . , , ,- , . , • ■ , cal subjects bejore this rule — giving precedence, for example, to the subjects of Percentage, In- in what they td'cst, Discount, Insurancc, &c. It is not easy consist. to discover the motive of this change. It is Ratio pnrt of certain that the proportion and ratio of num- the science. bers are parts of the science of arithmetic ; and Should pre- the properties of numbers which they unfold, cede applica- uons. are indispensably necessary to a clear apprehen- sion of the principles from which the practical rules are constructed. We may, it is true, explain each example m Percentage, Interest, Discount, Insurance, &:c., Cannot well by a Separate analysis. But this is a matter order. of much labor ; and besides, does not conduct the mind to any general principle, on which all the operations depend. Whereas, if the Rule of Three be explained, before entering on the Advantages practical subjccts, it is a great aid and a pow- ?.,'.^ „*'.'[,' erful auxiliary in explaininsr and establishing plaining tno J JT o a ^""^''f all the practical rules. If the Rule of Three Three. ^ is to be learned at all, should it not rather precede than follow its applications ? It is a great point, in instruction, to lay down a gen- The great gj.g^j principle, as early as possible, and then con- principleof ^ 1 ' J i instruction, nect witli it all subordinate operations. CHAP, ir.] >VRITH:.rETIC LANGUAGE. 183 ARITHMETICAL LANGUAGE, § 196. We have seen that the arithmetical al- Arithmetical ■ -n 1 alphabet. phabet contains ten characters.* From these elements the entire language is formed; and we now propose to show in how simple a manner. The names of the ten characters are the first Names of the cliaracters. ten words of the language. If the unit 1 be added to each of the numbers from to 9 in- First ten elusive, we find the first ten combinations in tions. arithmetic. t If 2 be added, in like manner, we have the second ten combinations ; adding Second ten, ami so on fof 3, gives us the third ten combinations ; and so othera. on, until we have reached one hundred com- binations (page 123). Now, as we progressed, each set of combina- Each setgiv. ing one addi- tions introduced one additional word, and the tionaiword. results of all the combinations are expressed by the words from two to twenty inclusive. § 197. These one hundred elementary com- ah that need be commit' binations, are all that need be committed to ted tome- memory ; for, every other is deduced from them, °^^' They are, in fact, but different spellings of the first nineteen words which follow one. If we ex- tend the words to one hundred, and recollect that * Section 114. f Section 116, 184 MATHEMATICAL SCIENCE. [bOOK II. at one hundred, we begin to repeat the numbers, worJstobe we See that we have but one hundred words to remembered i i r i i- • i r i ti for addition. 06 remembered lor addition; and ot these, alt Only ten obove ten are dericaiice. To this number, words primi- tive, must of course be added the few words which express the sums of the hundreds, thousands, &c. Subtraction: § 108, In Subtraction, we also find one hun- dred elementary combinations; the results of which are to be read.* These results, and all Number of the iiumbers employed in obtaining them, are words. expressed by twenty words. Muitipiica- § 199. In Multiplication (the table being car- ried to twelve), we have one hundred and forty- four elementary combinations,! and fifty-nine Number of Separate words (already known) to express the results of these combinations. Division: § 200. In Divisiou, also, we have one hundred and forty-four elementary combinations,! but Number of -^ •' ^ words. ygg only twelve words to express their results. Four hun- dred and o ^q-^^ T\\\xs, wc havc four hundred and eigh- ighty-eight " ' » elementary ty-cight elementary combinations. The results combina- tions, of these combinations are expressed by one hun- wordsused: j^,^^ words ; viz. nineteen in addition, ten in sub- 19 in addi- tion' traction, fifty-nine in multiplication, and twelve 10 in subtrac tion, — ■ 59inmuiti- * Section 127. f Section 129. :t Section 130. plication, ' ^ CHAP. II.] ARITHMETIC LANGUAGE. 185 in division. Of the nineteen words which are isin division employed to express the results of the combina- tions in addition, eight are again used to express similar results in subtraction. Of the fifty-nine which express the results of the combinations in multiplication, sixteen had been used to ex- press similar results m addition, and one in subtraction ; and the entire twelve, which ex- press the results of the combinations in division, had been used to express results of previous combinations. Hence, the results of all the ele- mentary combinations, in the four ground rules, are expressed by sixty-three different words ; and sixty-three I 11111 dilleiout they are the only words employed to translate words in aii. these results from the arithmetical into our com- mon language. The language for fractional units is similar Language , -P, r 1 '^s same for in every 'particular. By means oi a language fractions. thus formed we deduce every principle in the science of numbers. § 202. Expressing these ideas and their com- binations by figures, gives rise to the language Language of f. ., . -pi I •iri'i aiitlimetic: OI arithmetic. r5y the aid oi this language we not only unfold the principles of the science, its value and but are enabled to apply these principles to every question of a practical nature, involving the use of fio-ures. 186 MATHEMATICAL SCIENCE, [book II. But few combinations whicli change the sigaificatiun of the figures. Examples. Learn the language by use. Its grammar ; Alphabet — words, and their uses. § 203. There is but one further idea to be presented : it is this, — that there are very few combinations made amono; the fig;ures, which change, at all, their signification. Selecting any two of the figures, as 3 and 5, for example, we see at once that there are but three ways of writing them, that will at all change their signification. First, write them by the side of each ) 3 5, other ;5 3. Second, write them, the one ever i f, the other ) f- Third, place a decimal point before ^ .3, each ) .5. Now, each manner of writing gives a differ- ent signification to both the figures. Use, how- ever, has established that signification, and we know it, as soon as we have learned the lan- guage. We have thus explained what we mean by the arithmetical language. Its grammar em- braces the names of its elementary signs, or Alphabet, — the formation and number of its words, — and the laws by which figures are con- nected for the purpose of expressing ideas. We feel that there is simplicity and beauty in this system, and hope it may be useful. CHAP. II.] ARITHMETIC DEFINITIONS. 187 NECSSS:T7 of exact definitions and TERMS. § 204. The principles of every science are Frincipiesoi a collection of mental processes, having estab- lished connections with each other. In every branch of mathematics, the Definitions and Dcflnitions _ . 'u^d terms : Terms give form to, and are the signs of, cer- tain elementary ideas, which are the basis of the science. Between any term and the idea which it is employed to express, the connection should be so intimate, that the one will always suggest the other. These definitions and terms, when their sig- when once mtications are once hxed, must always be used always be in the same sense. The necessity of this is most same sense. urgent. For, "in the loJioIe range of arithmetical science there is no logical test of ti'ufh, but in Reason. a Cunformity of the reasoning to the definitions :nd terms, or to such princij)les as have been established from them." § 205. With these principles, as guides, we Definitions ^ , , ^ . . , and terms propose to examine some oi the dennitions and examined, terms which have, heretofore, formed the basis of the arithmetical science. We shall not con- fine our quotations to a single author, and shall ' make only those which fairly exhibit the gen- eral use of the terms, 188 MATHEMATICAL SCIENCE. [bOOK II. It is said, Number de " Numhev signifies a unit, or a collection of fliic-d. . . ,, units. How " The common method of expressing numbers is by the Arabic Notation. The Arabic method employs the following ten characters^, ov figures," &c. Names of the "The first nine are called significant figures, characters. i i i i i because each one always has a value, or denotes some number." And a little further on we have, Fisures have " The different values which figures have, are valma. called simple and local values." The definition of Number is clear and cor- Number rcct. It is a general term, comprehending al. fined: ^^^ phrascs which are used, to express, either separately or in connection, one or more things Also figures, of the samc kind. So, likewise, the definition of figures, that they are characters, is also right. Definition de- But mark how soon these definitions are de- parted from. The reason given why nine of the figures are called significant is, because " each one always has a value, or denotes some num- ber." This brings us directly to the question. Has a figure whether a figure has a value; or, whether it is a mere representatioe of value. Is it a number or a character to represent number ? Is it a It is merely . 7 7 -j t • i r^ i 1 7 a character: quantity ov symool f It is denned to be a char' rilAP. II.] ARITHMETIC DEFINITIONS, 189 acter which stands for, or expresses a number. Has it any other signification? How then can we say that it has a value — and how is it possi- iias novaiur. ble that it can have a simple and a local value ? The things which the figures stand for, may change their value, but not the figures them- selves. Indeed, it is very di/ficult for John to perceive how the figure 2, standing in the sec- but stands . . for value. ond place, is ten times as great as the same iig- ure 2 standing in the first place on the right! although he will readily understand, when the arithmetical language is explained to him, that the UNIT of one of these places is ten times as unit of place. great as that of the other. § 206. Let us now examine the leading defi- Leadi-ig deo nition or principle which forms the basis of the arithmetical lancruao-e. It is in these words : " Numbers increase from right to left in a of number. tenfold ratio ; that is, each removal of a figure one place towards the left, increases its value ten times." Now, it must be remembered, that number Does not has been defined as signifying "a unit, or a thedefini- collection of units." How, then, can it have a «"" "^^fo^ right hand, or a left ? and how can it increase from right to left in a tenfold ratio ?" The explanation given is, that "each removal of a 190 MATHEMATICAL SCIENCE. [bOOK II. Explanation. figuvB 0716 pluce toicavds the left, incrcases its value ten times." Number, signifying a collection of units, must Increase of iiecessarily increase according to the law by numbers has i • i i • i • i i i i noconneciion ■vvhich thcsc units are combmed ; and that law with figures. ^^ increase, whatever it may be, has not the slightest connection with the figures which are used to express the numbers. Ratio. Besides, is the term ratio (yet undefined), one which expresses an elementary idea? And "Tenfold is the term, a " tenfold ratio," one of sufficient simplicity for the basis of a system? Does, then, this definition, which in substance is used by most authors, involve and carry to Four leading the mind of the young learner, the four leading numberl^ idcas which form the basis of the arithmetical notation ? viz. : First. 1st. That numbers are expressions for one or more things of the same kind. Second. 2d. That numbers are expressed by certain characters called figures ; and of which there are ten. Third. 3d. That each figure always expresses as many units as its name imports, and no more. Fourth. 4th. That the Mnd of thing which a figure expresses depends on the place which the figure occupies, or on the value of the units, indicated in some other way. CHAP. II.] ARITHMETIC DEFINITIONS. 191 Place is merely one of the forms of language Place; by which we designate the unit of a number, itaosce. expressed by a figure. The definition attributes this property of place both to number and fig- ures, while it belongs to neither. § 207. Having considered the definitions and terms in the first division of Arithmetic, viz. in notation and numeration, we will now pass to Definitions in , , . * 1 1 • • Addition : the second, viz. Addition. The following are the definitions of Addition, taken from three standard works before me : " The putting together of two or more num- First bers (as in the foregoing examples), so as to make one whole number, is called Addition, and the whole number is called the sum, or amount." ' " Addition is the collecting of numbers to- second, gether to find their sum." " The process of uniting two or more num- Third. hers together, so as to form one single number, IS called Addition." " The answer, or the number thus found, is called the sum, or amount." Now, is there in either of these definitions Defects, any test, or means of determining when the pupil gets the thing he seeks for, viz. " the sum of two or more numbers ?" A^o previous defi- Reaaoa nition has been given, in either work, of the 192 MATHEMATICAL SCIENCE. [liUOKll term sum. How is the learner to know, what he is seeking for, unless that thing be defined? Noprin- Suppose that John be required to find the sum ciple as a standard. 01 the numbers 3 and 5, and pronounces it to be 10. How will you correct him, by showing that he has not conformed to the definitions and rules ? You certainly cannot, because you have established no test of a correct process. But, if you have previously defined sum to be a number which contains as many units as there are in all the numbers added : or, if you say, Correct defl- "Addition is the process of uniting two or more numbers, in such a way, that all the units . which they contain may be expressed by a sin- gle number, called the sum, or sum total ;" you * will then have a test for the correctness of the civesatest. pi'ocess of Addition; viz. Does the number, which you call the sum, contain as many units as there are in all the numbers added ? The answer to this question will show that John is wrong. Deiinitions jf § 308. I wiU now quote the definitions of Fractions from the same authors, and in the same order of reference. Fii-at. " We have seen, that numbers expressing whole things, are called integers, or whole numbers ; but that, in division, it is often necessary to CHAP, II.] ARITHMETIC EEFINITIONl 193 Second. Tliiid. Tenn fraction defined. Ideas divid". or break a whole thing into parts, and that these parts are called fractions, or broken numbers." " Fractions are parts of an integer." " When a number or thing is divided into equal parts, these parts are called Fractions." Now, will either of these definitions convey to the mind of a learner, a distinct and exact idea of a fraction ? The term " fraction," as used in Arithmetic, means one or more equal parts of something regarded as a whole : the parts to be expressed in terms of the thing divided considered as a UNIT. There are three prominent ideas which the mind must embrace : 1st. That the thing divided be regarded as a standard, or unity ; 2d. That it be divided into equal parts ; 3d. That the parts be expressed in terms of the thing divided, regarded as a unit. These ideas are referred to in the latter part of the first definition. Indeed, the definition would suggest them to any one acquainted with the subject, but not, we think, to a learner. In the second definition, neither of them is hinted at. Take, for example, the integer num- ber 12, and no one would say that any one part of this number, as 2, 4, or 6, is a fraction. 13 First. Second, Tliird. The defini- tions exam ined: Is a frac- tion part of an integer 194 MATHEMATICAL SCIENCE. [bOOK 11 Third The third definition would be perfectly accu- definition ; . . r i i i • m i rate, by inserting alter the word " thing, the words, " regarded as a whole." It very clearly expresses the idea of equal parts, but does not In what de- present the idea strongly enough, that the thing divided must be regarded as unity, and that the parts must be expressed in terms of this unity. § 209. I have thus given a few examples, illus- Necessity of ti'atiug the necessity of accurate definitions and terms. Nothing further need be added, except the remark, that terms should always be used in the same sense, precisely, in which they are de- fined. Objection To some, perhaps, these distinctions may ap- of thought P6^r over-nice, and matters of little moment. nnd language, j^ ^^^ ^^ supposcd that a general impression, imparted by a language reasonably accurate, will suffice very well ; and that it is hardly worth while to pause and weigh words on a nicely-adjusted balance. Any such notions, permit me to say, will lead to fatal errors in education. Definitions in It is iu mathematical science alone that words ''"™'^' ■ are the signs of exact and clearly-defined ideas. It is here only that we can see, as it were, the very thoughts through the transparent words by which they are expressed. If the words of the CHAP. II.] ARITHMETIC SUBJECTS. 196 reason cur redly. definitions are not such as convey to the mind Must be GXflCt to of the learner, the fundamental ideas of the science, he cannot reason upon these ideas ; for, he does not apprehend them ; and the great reasoning faculty, by which all the subsequent principles of mathematics are developed, is en- tirely unexercised.* It is not possible to cultivate the habit of camuii other- .... . , . wise cultivate accurate thmking, without the aid and use of habits of exact language. No mental habit is more use- ful than that of tracing out the connection be- tween ideas and language. In Arithmetic, that connection can be made strikingly apparent. Connection Clear, distinct ideas — diamond thoughts — may worJsand be strung through the mind on the thread of ""'^^^'^'*.« ~ o anthinetic. science, and each have its word or phrase by which it can be transferred to the minds of others. now SHOULD THE SUBJECTS BE PRESENTED ? § 210. Having considered the natural connec- vvhai lion of the subjects of arithmetic with each considmi