IN MEMORIAM FLORIAN CAJORI THE LOGIC AND UTILITY OF MATHEMATICS, WITH THE BEST METHODS OF INSTRUCTION" EXPLAINED AND ILLUSTRATED. BY CHARLES DAYIES, LL.D. NEW YORK: PUBLISHED BY A. S. BARNES & CO. NO. 51 JOHN-STEEET. CINCINNATI : H. W. DERBY & COMPANY. 1851. Entered according to Act of Congress, in the year Eighteen Hundred and fifty, BY CHARLES DAVIES, In the Clerk s Office of the District Court of the United States for the Southern District of New York. STEREOTYPED BY BICHARD C. VALENTINE, NEW YORK. F. C. GUTIERREZ, Printer, No. 51 John-street, corner of Dutch. PREFACE. THE following work is not a series of speculations. It is but an analysis of that system of mathematical instruction which has been steadily pursued at the Military Academy over a quarter of a century, and which has given to that institution its celebrity as a school of mathematical science. It is of the essence of that system that a principle be taught before it is applied to practice ; that general principles and gen eral laws be taught, for their contemplation is far more improving to the mind than the examination of isolated propositions ; and that when such principles and such laws are fully compre hended, their applications be then taught as consequences or practical results. This view of education led, at an early day, to the union of the French and English systems of mathematics. By this union the exact and beautiful methods of generalization, which distinguish the French school, were blended with the practical methods of the English system. The fruits of this new system of instruction have been abun dant. The graduates of the Military Academy have been sought for wherever science of the highest grade has been PREFACE. needed. Russia has sought them to construct her railroads ;* the Coast Survey needed their aid ; the works of internal im provement of the first class in our country, have mostly been conducted under their direction ; and the recent war with Mexico afforded ample opportunity for showing the thousand ways in which science the highest class of knowledge may be made available in practice. All these results are due to the system of instruction. In that system Mathematics is the basis Science precedes Art Theory goes before Practice the general formula embraces all the particulars. It was deemed necessary to the full development of the plan of the work, to give a general view of the subject of Logic. The materials of Book I. have been drawn, mainly, from the works of Archbishop Whately and Mr. Mill. Although the general outline of the subject has but little resemblance to the work of either author, yet very much has been taken from both ; and in all cases where it could be done consistently with my own plan, I have adopted their exact language. This remark is par ticularly applicable to Chapter III., Book L, which is taken, with few alterations, from Whately. For a full account of the objects and plan of the work, the reader is referred to the Introduction. FISHKILL LANDING, June, 1850. * Major Whistler, the engineer, to whom was intrusted the great enterprise of constructing a railroad from St. Petersburg to Moscow, and Major Brown, who succeeded him at his death, were both graduates of the Military Acad emy. CONTENTS INTRODUCTION. PAGE OBJECTS AND PLAN OF THE WORK 11 BOOK I, LOGIC. CHAPTER L DEFINITIONS OPERATIONS OF THE MIND TERMS DEFINED. . 27 SECTION Definitions 1 6 Operations of the Mind concerned in Reasoning 6 12 Abstraction 1214 Generalization 14 22 Terms Singular Terms Common Terms 15 Classification 16 20 Nature of Common Terms 20 Science 21 Art . 22 CONTENTS. CHAPTER II. PAGE SOURCES AND MEANS OF KNOWLEDGE INDUCTION 41 SECTION Knowledge 23 Facts and Truths 24 27 Intuitive Truths 27 Logical Truths 28 Logic 29 Induction . 3034 CHAPTER III. DEDUCTION NATURE OF THE SYLLOGISM ITS USES AND AP PLICATIONS Page 54 SECTION Deduction 34 Propositions 3540 Syllogism 40 42 Analytical Outline of Deduction 4267 Aristotle s Dictum 54 61 Distribution and Non-distribution of Terms 61 67 Rules for examining Syllogisms 67 Of Fallacies 68 71 Concluding Remarks 71 75 CONTENTS. BOOK II, MATHEMATICAL SCIENCE. CHAPTER I. QUANTITY AND MATHEMATICAL SCIENCE DEFINED DIFFER ENT KINDS OF QUANTITY LANGUAGE OF MATHEMATICS EXPLAINED SUBJECTS CLASSIFIED UNIT OF MEASURE DEFINED MATHEMATICS A DEDUCTIVE SCIENCE . . . .Page 99 SECTION Quantity 75 79 Number . 7981 Space 8187 Analysis 87 91 Language of Mathematics 91 94 Quantity Measured 9497 Pure Mathematics 97-101 Comparison of Quantities 101 Axioms or Formulas for inferring Equality 102 Axioms or Formulas for inferring Inequality 102 CHAPTER II. PAGB ARITHMETIC SCIENCE AND ART OF NUMBERS 117 SECTION I. SECTION First Notions of Numbers 104107 Ideas of Numbers Generalized 107 110 Unity and a Unit Defined 110 Simple and Denominate Numbers Ill 113 Alphabet Words Grammar 113 Arithmetical Alphabet 114 Spelling and Reading in Addition 115 120 Spelling and Reading in Subtraction 120 122 CONTENTS. SECTION Spelling and Reading in Multiplication 122 Spelling and Reading in Division 123 Units increasing by the Scale of Tens 124131 Units increasing by Varying Scales 131 Integer Units of Arithmetic 132 Abstract or Simple Units 132 134 Units of Currency 134136 Units of Weight 136139 Units of Measure 139 150 Advantages of the System of Unities 150 System of Unities applied to the Four Ground Rules. 151 155 SECTION II. Fractional Units changing by the Scale of Tens 155158 Fractional Units in general 158 161 Advantages of the System of Fractional Units 161163 SECTION III. Proportion and Ratio 163 172 SECTION IY. Applications of the Science of Arithmetic 172 180 SECTION Y. Methods of teaching Arithmetic considered 180 Order of the Subjects 180 183 1st. Integer Units 183185 2d. Fractional Units 185 3d. Comparison of Numbers, or Rule of Three 186188 4th. Practical Part, or Applications of Arithmetic 188 Objections to Classification answered 189 191 Objections to the new Method 191 Arithmetical Language 192 200 Necessity of exact Definitions and Terms 200206 How should the Subjects be presented 206209 Text-Books 209214 First Arithmetic 214 227 CONTENTS. SECTION Second Arithmetic 227231 Third Arithmetic 231236 Concluding Remarks 236 CHAPTER III. GEOMETRY DEFINED THINGS OF WHICH IT TREATS COM PARISON AND PROPERTIES OF FIGURES DEMONSTRATION PROPORTION SUGGESTIONS FOR TEACHING .... Page 223 SECTION Geometry 237 Things of which it treats 238248 Comparison of Figures with Units of Measure 249 256 Properties of Figures 256 Marks of what may be proved 257 Demonstration 258267 Proportion of Figures 267270 Comparison of Figures 270 273 Recapitulation Suggestions for Teachers 273 CHAPTER IV. ANALYSIS ALGEBRA ANALYTICAL GEOMETRY DIFFEREN TIAL AND INTEGRAL CALCULUS Page 261 SECTION Analysis 274280 Algebra 280 Analytical Geometry 281283 Differential and Integral Calculus 283 286 Algebra further considered 286296 Minus Sign 296298 Subtraction 298 Multiplication 299302 Zero and Infinity 302307 Of the Equation 307 31 1 Axioms 311 Equality its meaning in Geometry 312 Suggestions for those who teach Algebra 315 L 10 CONTENTS. BOOK III, UTILITY OF MATHEMATICS. CHAPTER I. THE UTILITY OF MATHEMATICS CONSIDERED AS A MEANS OF INTELLECTUAL TRAINING AND CULTURE . . Page 293 O CHAPTER II. THE UTILITY OF MATHEMATICS REGARDED AS A MEANS OF ACQUIRING KNOWLEDGE BACONIAN PHILOSOPHY 308 CHAPTER III. THE UTILITY OF MATHEMATICS CONSIDERED AS FURNISHING THOSE RULES OF ART WHICH MAKE KNOWLEDGE PRACTI CALLY EFFECTIVE . . 325 APPENDIX. A COURSE OF MATHEMATICS WHAT IT SHOULD BE 341 ALPHABETICAL INDEX 353 INTRODUCTION OBJECTS AND PLAN OF THE WORK. UTILITY and Progress are the two leading utility ideas of the present age. They were manifested Progress: in the formation of our political and social insti- Their influ- Gncc in ffov* tutions, and have been further developed in the eminent: extension of those institutions, with their subdu ing and civilizing influences, over the fairest por tions of a great continent. They are now be coming the controlling elements in our systems in education, of public instruction. What, then, must be the basis of that system what the basis of of education which shall embrace within its ho- utility and rizon a Utility as comprehensive and a Progress as permanent as the ordinations of Providence, exhibited in the laws of nature, as made known by science? It must obviously be laid in the examination and analysis of those laws ; and 12 INTRODUCTION. . primarily; in those preparatory studies which fit ^ ? ; : % arid . qualify trie mind for such Divine Contem plations. Bacon s When Bacon had analyzed the philosophy of Philosophy. the ancients, he found it speculative. The great highways of life had been deserted. Nature, spread out to the intelligence of man, in all the minuteness and generality of its laws in all the harmony and beauty which those laws develop had scarcely been consulted by the ancient phi- Phiioso- losophers. They had looked within, and not phy of the Ancients, without. They sought to rear systems on the uncertain foundations of human hypothesis and speculation, instead of resting them on the im mutable laws of Providence, as manifested in the material world. Bacon broke the bars of this mental prison-house : bade the mind go free, and investigate nature. Foundations Bacon laid the foundations of his philosophy in Phiklophy : or g arnc l aws > an d explained the several processes of experience, observation, experiment, and in duction, by which these laws are made known, why op- He rejected the reasonings of Aristotle because posed to Aris totle s, they were not progressive and useful ; because they added little to knowledge, and contributed nothing to ameliorate the sufferings and elevate the condition of humanity. PLAN OF THE WORK. 13 The time seems now to be at hand when the Practical philosophy of Bacon is to find its full develop ment. The only fear is, that in passing from a speculative to a practical philosophy, we may, for a time, lose sight of the fact, that Practice without Science is Empiricism; and that all its true na- which is truly great in the practical must be the application and result of an antecedent ideal. What, then, are the sources of that Utility, what is and the basis of that Practical, which the pres- temofedu- ent generation desire, and after which they are so anxiously seeking ? What system of training and discipline will best develop and steady the intellect of the young ; give vigor and expan sion to thought, and stability to action ? What which win course of study will most enlarge the sphere of steady the investigation ; give the greatest freedom to the mind without licentiousness, and the greatest freedom to action consistent with the laws of nature, and the obligations of the social com pact ? What subject of study is, from its na- what are . . the subjects ture, most likely to ensure this training, and contribute to such results, and at the same time lay the foundations of all that is truly great in the Practical ? It has seemed to me that math- Mathematics. ematical science may lay claim to this pre-emi nence. 14 INTRODUCTION. The first impressions which the child receives of Number and Quantity are the foundations of knowledge, j^ mathematical knowledge. They form, as it were, a part of his intellectual being. The laws Laws of o f Nature are merely truths or generalized facts, Nature. . . in regard to matter, derived by induction from experience, observation, and experiment. The laws of mathematical science are generalized Number truths derived from the consideration of Number g a ^ and Space. All the processes of inquiry and investigation are conducted according to fixed laws, and form a science ; and every new thought and higher impression form additional links in the lengthening chain. Mathemat- The knowledge which mathematical science ical knowl- . edge . imparts to the mind is deep profound abiding. It gives rise to trains of thought, which are born in the pure ideal, and fed and nurtured by an acquaintance with physical nature in all its mi- what it nuteness and in all its grandeur : which survey does. the laws of elementary organization, by the mi croscope, and weigh the spheres in the balance of universal gravitation. What The processes of mathematical science serve to give mental unity and wholeness. They im part that knowledge which applies the means of PLAN OF THE WORK. 15 crystallization to a chaos of scattered particulars, Right knowi- and discovers at once the general law, if there tL^eanlTof be one, which forms a connecting link between cry t ^ liza " them. Such results can only be attained by minds highly disciplined by scientific combina tions. In all these processes no fact of science is forgotten or lost. They are all engraved on the great tablet of universal truth, there to be read by succeeding generations so long as the it records . ., . . . and preserves laws of mind remain unchanged. This is stri- trut h. kingly illustrated by the fact, that any diligent student of a college may now read the works of Newton, or the Mecanique Celeste of La Place. The educator regards mathematical science HOW the educator re- as the great means of accomplishing his work, gardsmatn- The definitions present clear and separate ideas, which the mind readily apprehends. The axioms The axioms. are the simplest exercises of the reasoning fac ulty, and afford the most satisfactory results in the early use and employment of that faculty. The trains of reasoning which follow are com binations, according to logical rules, of what has been previously fully comprehended; and influence of , the study of the mind and the argument grow together, so mat hematica that the thread of science and the warp of the onthemind - intellect entwine themselves, and become insep arable. Such a training will lay the foundations 16 INTRODUCTION. of systematic knowledge, so greatly preferable to conjectural judgments. HOW the The philosopher regards mathematical science P regTrds** as tne mere tools of his higher Vocation. Look- mathematics: ^ w - t ^ a stea( jy anc [ an xious eye to Nature, and the great laws which regulate and govern all things, he becomes earnestly intent on their examination, and absorbed in the wonderful har monies which he discovers. Urged forward by its necessity these high impulses, he sometimes neglects that to him. thorough preparation, in mathematical science, necessary to success; and is not unfrequently obliged, like Antaeus, to touch again his mother earth, in order to renew his strength. The views The mere practical man regards with favor of the practi cal man. only the results of science, deeming the reason ings through which these results are arrived at, quite superfluous. Such should remember that instruments the mind requires instruments as well as the of the mind. hands, and that it should be equally trained in their combinations and uses. Such is, indeed, now the complication of human affairs, that to do one thing well, it is necessary to know the properties and relations of many things. Every Everything thing, whether existing in the abstract or in the has a law. material world ; whether an element of knowl- PL AN OF THE WORK. 17 edge or a rule of art, has its connections and its TO know ... the law is to law: to understand these connections and that knowt he law, is to know the thing. When the principle is clearly apprehended, the practice is easy. With these general views, and under a firm Mathematics conviction that mathematical science must be come the great basis of education, I have be stowed much time and labor on its analysis, as a subject of knowledge. I have endeavored to present its elements separately, and in their con- HOW. nections ; to point out and note the mental fac ulties which it calls into exercise ; to show why and how it develops those faculties; and in what respect it gives to the whole mental machinery greater power and certainty of action than can be attained by other studies. To accomplish what was deemed ne- these ends, in the way that seemed to me most cessary. desirable, I have divided the work into three parts, arranged under the heads of Book I., II., and III. Book I. treats of Logic, both as a science and Logic, an art ; that is, it explains the laws which gov ern the reasoning faculty, in the complicated processes of argumentation, and lays down the Explanation, rules, deduced from those laws, for conducting such processes. It being one of the leading 2 18 INTRODUCTION. For what objects to show that mathematical science is the sed best subject for the development and application of the principles of logic ; and, indeed, that the science itself is but the application of those prin- The necessity ciples to the abstract quantities Number and of treating it. . I Space, it appeared indispensable to give, in a manner best adapted to my purpose, an out line of the nature of that reasoning by means of which all inferred knowledge is acquired. Book ii. Book II. treats of Mathematical Science. Here I have endeavored to explain the nature of or what it the subjects with which mathematical science is treats. 1-1 i i conversant ; the ideas which arise in examining and contemplating those subjects ; the language employed to express those ideas, and the laws of their connection. This, of course, led to a class- Manner of ification of the subjects; to an analysis of the treating. language used, and an examination of the reason ings employed in the methods of proof. Book m. Book III. explains and illustrates the Utility of Utility of ,, -TV Mathematics. Mathematics : First, as a means of mental disci pline and training ; Secondly, as a means of ac quiring knowledge ; and, Thirdly, as furnishing those rules of art, which make knowledge prac- acally effective. PLANOFTHEWORK. 19 Having thus given the general outlines of the classes of work, we will refer to the classes of readers for whose use it is designed, and the particular ad vantages and benefits which each class may re ceive from its perusal and study. There are four classes of readers, who may, Four classes it is supposed, be profited, more or less, by the perusal of this work : 1st. The general reader ; First class. 2d. Professional men and students ; second. 3d. Students of mathematics and philosophy ; Third. 4th. Professional Teachers. Fourth. First. The general reader, who reads for im. Advantages provement, and desires to acquire knowledge, era i rea der. must carefully search out the import of language. He must early establish and carefully cultivate the habit of noting the connection between ideas connec- and their signs, and also the relation of ideas to ^ordsTud" each other. Such analysis leads to attentive reading, to clear apprehension, deep reflection, and soon to generalization. Logic considers the forms in which truth must Logic, be expressed, and lays down rules for reducing all trains of thought to such known forms. This habit of analyzing arms us with tests by which its value: we separate argument from sophistry truth from falsehood. The application of these principles, INTRODUCTION. in the study in the construction of the mathematical science, mathematics where the relation between the sign (or language) and the thing signified (or idea expressed), is un mistakable, gives precision and accuracy, leads to right arrangement and classification, and thus prepares the mind for the reception of general knowledge. Advantages Secondly. The increase of knowledge carries professio a) men. with it the necessity of classification. A limited number of isolated facts may be remembered, or a few simple principles applied, without tracing out their connections, or determining the places which they occupy in the science of general knowledge. But when these facts and principles are greatly multiplied, as they are in the learned The reason, professions ; when the labors of preceding gen erations are to be examined, analyzed, compared ; when new systems are to be formed, combining all that is valuable in the past with the stimu lating elements of the present, there is occasion for the constant exercise of our highest facul- ties. Knowledge reduced to order ; that is, knowledge so classified and arranged as to be eas i}y 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. Masses of facts, like masses of matter, are ca- Knowledge pable of very minute subdivisions ; and when we duced to te know the law of combination, they are readily elements - 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 Logic and ,. , , 11 i i ^ i mathematics. applicable to all kinds of 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 w r hich characterize his particular department of knowledge. Thirdly. Mathematical knowledge differs from Mathemati- every other kind of knowledge in this : it is, as ca ed n w it were, 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 extend the figure, I may add, that the at the right web of the spider, 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- cai science is acting, and to begin at the right place and pro ceed in the right 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 tempted, 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 . the whole about which mathematical science is conversant, the whole field to be gone over is at once sur veyed: by calling attention to the faculties of of^ntdTr! the m md which the science brin g s into exercise, ing the men- we are better prepared to note the intellectual tal faculties : operations which the processes require ; and by PLANOFTHEWORK. 23 a knowledge of the laws of reasoning, and an ofaknowi- ., edge of the acquaintance with the tests of truth, we are en- i awsofrea . abled to verify all our results. These means have been furnished in the following work, and to aid the student in classification and arrangement, diagrams have been prepared exhibiting separ- what has . been done. ately and m connection all the principal parts ot mathematical science. The student, therefore, who adopts the system here indicated, will find his way clearly marked out, and will recognise, Advanta ^ e3 J J to the stu- from their general resemblance to the descrip- <ient. tions, 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- Where 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 rder 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 ,,.,,. . , . the subject. and lifting the mind towards the same eminence. Fourthly. The leading idea, in the construe- Advantages tion of the work, has been, to afford substantial professional aid to the professional teacher. The nature of teacher - 24 INTRODUCTION. His duties: his duties their inherent difficulties the per- plexities which meet him at every step the want ments and . , . , ( . ,. difficulties: oi sympathy and support in his hours 01 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 Remoteness life, spends his days in raising the feeble mind from active ,. , . life . or childhood to strength in planting aright the seeds of knowledge in curbing 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 his efforts, he is certain that his labors have not been in 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 his hands the keys of knowledge If tance of his , labors. tne nrst 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. zon of civilization. He impresses himself on The influence r , . i i i T i f h s labors. 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 sources of 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 Ob J ectsfor which the analysis of mathematical science. I have spread work was undertaken. out, in detail, those methods which nave 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 ; for, the principles of imparting instruction are the same for all branches of knowledge. The system which I have indicated is com plete in itself. It lays open to the teacher the entire skeleton of the science exhibits all its parts separately and in their connection. It explains a course of reasoning simple in itself, and applicable not only to every process in mathematical science, but to all processes of argumentation in every subject of knowledge. The teacher who thus combines science with art, no longer regards Arithmetic as a mere treadmill of mechanical labor, but as a means System. What it presents. What it explains. Science combined with art: 26 INTRODUCTION. and the simplest means of teaching the art and tages result- . - . . , , . ing from it. science of reasoning on quantity and this is the logic of mathematics. If he would accom- Resuitsof plish well his work, he must so instruct his right instruc tion, pupils that they shall apprehend clearly, think quickly and correctly, reason justly, and open their minds freely to the reception of all knowl edge. BOOK I. LOGIC, CHAPTER I. DEFINITIONS - OPERATIONS OF THE MIND - TERMS DEFINED. DEFINITIONS. 1. DEFINITION is a metaphorical word, which Definition 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 ^ meaning of the word ; while in others, besides , others imply explaining the meaning of the word, it is also tllings implied that there exists, or may exist, a thing corresponding to the word. words - 2. Definitions which do not imply the exist- or definitions ... which do ence of things corresponding to the words de- not imply fined, are those usually found in the Dictionary of one s own language. They explain only the to words. 28 LOGIC. [BOOK i. Ther meaning of the word or term, by giving some explain equivalent expression which may happen to be words by equivalents, better known. Definitions which imply the ex istence of things corresponding to true words de fined, do more than this. Definition p O r example : " A triangle is a rectilineal fig- of a triangle; ure having three sides." This definition does w ! t iat two things : implies. i$t 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. o fa 3. To define a word when the definition is definition which im- to imply the existence of a thing, is to select, plies the ex- / u i r i i isuaice of " om a ^ the properties of the thing those which a thmg. are most s i m p] 6j genera^ anc [ obvious ; and the Properties properties must be very well known to us before must be . known. we can 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 supports . tru th. kind of definition is not only the best form of ex pressing certain conceptions, but also contributes to the development and support of new truths. In 4. In Mathematics, and indeed, in all strict Mathematics names imply sciences, names imply the existence of the things CHAP. I.] DEFINITIONS. 29 things and express attributes. which they name ; and the definitions of those names express attributes of the things ; so that no correct definition whatever, of 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 Definition3 is a tacit assumption of some proposition which ofthisclas3 is expressed by means of the definition, and propositions, which gives to such definition its importance. 5. All the reasonings in mathematics, which Reasoning rest ultimately on definitions, do, in fact, rest restingon J definitions ; on the intuitive inference, that things corre- rests on spending 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 tions. 1st. The definition must be adequate : that is, neither too extended, nor too narrow for the word defined. 2d. The definition must be in itself plainer than the word defined, else it would not explain it. 3d. The definition should be expressed in a convenient number of appropriate words. 4th. When the definition implies the existence of a thing corresponding to the word defined, the certainty of that existence must be intuitive. Four rules. 1st rule. 2cl rule. 3d rule. 4th rule. 30 LOGIC. [BOOK i. OPERATIONS OF THE MIND CONCERNED IN REASONING. Three opera- g. There are three operations of the mind tions of the which are immediately concerned in reasoning. 1st. Simple apprehension ; 2d. Judgment ; 3d: Reasoning or Discourse. dim lea $ 7- Simple apprehension is the notion (or prehension, conception) of an object in the mind, analogous to the perception of the senses. It is either incompiex. Incomplex 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 : Complex is of several with such a relation, as of a triangle within a circle, or a circle within a square. 8. Judgment is the comparing together in the mind two of the notions (or ideas) which Judgment defined, are the objects of apprehension, whether com plex or incompiex, and pronouncing that they agree or disagree with each other, or that one of them belongs or does not belong to, the other : for example : that a right-angled triangle and an Judgment equilateral triangle belong to the class of figures either called triangles ; or that a square is not a circle. or l Judgment, therefore, is either Affirmative or Neg- ncgative - ative. CHAP. I.] ABSTRACTION. 31 9. Reasoning (or discourse) is the act of reasoning proceeding from certain judgments to another founded upon them (or the result of them). 10. Language affords the signs by which Language these operations of the mind are recorded, ex- 8ignsof pressed, and communicated. It is also an in- thou s nt: 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 has its own and for a mind not previously versed in the language, meaning and right use of the various words and which signs which constitute the language, to attempt must be learned. the study of methods of philosophizing, would be as absurd as to attempt reading before learn ing the alphabet. ABSTRACTION. 12. The faculty of abstraction is that power f. , i , . Abstraction, ol the mind which enables us, in contemplating any object (or objects), to attend exclusively to 32 LOGIC. [BOOK i. some particular circumstance belonging to it, and quite withhold our attention from the rest. Thus, in contempia- if a person 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 ; the process or 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 The term denote tne operation of abstracting from one or Abstraction, mO re things the particular part under consider- how used. ation ; and likewise to designate the state of the mind when occupied by abstract ideas. Hence, abstraction is used in three senses : Abstraction lst - To denote a faculty or power of the 2d - To denote a process of the mind ; and, GENERALIZATION. Generalize l4 Generalization is the process of con templating the agreement of several objects in certain points (that is, abstracting the circum stances of agreement, disregarding the differ- CHAP. I.] TERMS. 33 ences), and giving to all and each of these ob- ofseveral jects a name applicable to them in respect to thin g 8 - this agreement. For example ; we give the name of triangle, to every rectilineal figure hav ing three sides : thus we abstract this property from all the others (for, the triangle has three angles, may be equilateral, or scalene, or right- angled), and name the entire class from the prop erty so abstracted. Generalization therefore necessarily implies abstraction ; though abstrac tion does not imply generalization. Generaliza tion implies abstraction. A term. TERMS - SINGULAR TERMS - COMMON TERMS. 15. An act of apprehension, expressed in language, is called a Term. Proper names, or 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 common 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 Significates. 3 terms. 34 LOGIC. [BOOK i. CLASSIFICATION. 6 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 Genus, called a Genus or Species genus being the Bpecies. more extensive term, and often embracing many species. Exam lea ^ or exam pl e : animal is a genus embracing in every thing which is endowed with life, the pow- classification. er of voluntary motion, and sensation. It has many species, such as man, beast, bird, &c. If we say of an animal, that it is rational, it !><- 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 specLs bird. A species may likewise be divided into classes, Subspecies or or subspecies ; thus the species man, may be divided into the classes, male and female, tind these classes may be again divided until we reach the individuals. Principles 17. Now, it will appear from the principles classification, which govern this system of classification, that CHAP. I.] CLASSIFICATION. 35 the characteristic of a genus is of a more exten- Genus more . ~ , , ,, extensive sive signification, but involves fewer particu- than species, lars than that ol a species. In like manner, the characteristic of a species is more extensive, but less full and complete, than that of a subspecies but less ful1 and 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 / i\ 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 / 2 \ 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 i. and 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 quadn- vfduti famng lateral; and hence, possesses the characteristic under the ^ ^ genus ; and each variety of both species genus enjoys aii 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. 18. In special classifications, it is often not necessary to begin with the most general char- _ . .. acteristics; and then the genus with which we Subaltern 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. genus. Parallelo gram. Highest 19. A genus which cannot be considered 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 . . genus. 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 COMMON TERMS. $ 20. It should be steadily kept in mind, that the "common terms" employed in classification, Acommon have not, as the names of individuals have, any termh < 18 * no real thing real existing thing in nature corresponding to corres Pnd- 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 differs from others of the same class, it thing in is applicable equally well to all or any of them. ind * vi d ualg Tims, quadrilateral denotes no real thing, dis- dLffer; 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 J but is this means, a common term becomes applicable applicable to many 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 specting the results of this process : many hav- difficult y- ing contended, and perhaps more having taken 38 LOGIC. [BOOK i. the interpre tation of common terms. Difficulty in it for granted, that there must be some really existing thing corresponding to each of those 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 smgu | ar name yEtna," it has been supposed correspond- o in<? to each. fa a t fa e 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) inadequate n ti n of an individual ; and from the very cir- on par- cums f; ance o f its inadequacy, it will apply equally signaling we j} ^o a ny one of several individuals. For ex- the thing. ample ; if I omit the mention and the consider ation of every circumstance which distinguishes yEtna from any other mountain, I then form ^^ notion, that inadequately designates /Etna. This u Mountain" notion is expressed by the common term " moun- tam " which does not imply any of the peculiar- toa11 ities of the mountain /Etna, and is equally ap- moun tains. 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. I.] SCIENCE. 39 characteristic which we choose to abstract and May fix on attributes consider as the basis of our classification, disre- arbitrarily garding all the rest : so that the same individual classi k c r . ltion 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- Ha sa vantageously applied. In a more strict and g^oition. technical sense, it has another signification. " Every thing in nature, as well in the in- yiewg of animate as in the animated world, happens or Kaut - 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- Generallaws - 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 want of rule. known to us. f Assuming that all the phenomena of nature * Section 23. f Kant. 40 LOGIC. [BOOK i. science are consequences of general and immutable laws, a technical we may define Science to be the analysis of sense defined: ^^ j awg> comprehending not only the con- an analysis nec t e d processes of experiment and reasoning of the laws 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. ART. 22. Art is the application of knowledge to Arti practice. Science is conversant about knowl- appucation e( }g e . Art is the use or application of knowl- 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 plied, becomes a rule of art. The developments of science increase knowledge : the applications and of art add to works. Art, necessarily, presup- prosupposea p 0ses knowledge : art, in any but its infant state, knowledge. r J 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 of a single art. Such is the complication of hu- Many things . must be man anairs, that to enable one thing to be done, ibreTn^ctn il: is often rec l uisite to know the nature and prop- be done, erties of many things. CHAP. II.] KNOWLEDGE. 41 CHAPTER II. SOURCES AND MEANS OF KNOWLEDGE INDUCTION. KNO WLEGDE. 23. KNOWLEDGE is a clear and certain con- Knowledge a clear con ception of that which is true, and implies three cep tionof things: what is true: 1st. Firm belief; 2d. Of what is true; and, implies ~, . , 1st. Firm 3d. On sufficient grounds. belief . If any one, for example, is in doubt respecting *&. of what is true ; one of Legendre s Demonstrations, he cannot 3d. on be said to know the proposition proved by -it. If, roim ^ again, he is fully convinced of any thing that is not true, he is mistaken in supposing himself to Examples. know it; and lastly, if two persons are each/^% confident, 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. 42 LOGIC. [BOOK I. FACTS AND TRUTHS. Knowledge is 24 UT knowledge is of two kinds : of facts of facts and and trut h s . A fact is any thing that HAS BEEN truths. J or is. That the sun rose yesterday, is a fact : that he gives light to-day, is a fact. That wa ter is fluid and stone solid, 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 shaii be. oi ascertaining truth : ofanrtri^ lst % comparing known facts with each ing 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 elements of all our knowledge, and that knowl edge itself is but their clear apprehension, their knowledge, firm 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 arrive at a perceive that it is shaken by the wind, and note knowledge of the fact that it is in motion. We decompose water 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 ways above indicated, does not point out any connec- This does not point out a tion between them. It merely exhibits them to connection the mind as separate or isolated; that is, each b t ^ en 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, nectiou that , . c . , -, , i ,j . is pointed out that inference is pointed out by the reasoning b y therea . process, which is carried on, in all cases, by com parison. 2dly. A result obtained by comparing facts, we Truth, found have designated by the term Truth. Truths, therefore, are inferences from facts ; and every * Under this term we include all the methods of inves tigation and processes of arriving at facts, except the pro cess of reasoning. 44 LOGIC. [BOOK i. and truth has reference to all the singular facts from inferred from them. 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. HOW 26. How, then, are truths to be inferred truths **--,. inferred from i rom tacts by the reasoning process? There facts by the reasoning process. j s ^ When the instances are so few and simple ist case, 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. INTUITIVE TRUTH. 27. Truths which become known by con- sidering all the facts on which they depend, and which are inferred the moment the facts are truths, apprehended, are the subjects of Intuition, and are called Intuitive or Self-evident Truths. The intuition ter m 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, and apprehend the relations of those facts at the same time, and by the same act by which we apprehend the facts themselves. Hence, intuitive or self-evi- HOW intuitive ... truths are 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 intervention of other ideas. They are necessary consequences of conceptions respecting which they are asserted. Axioms of The axioms of Geometry afford the simplest and Je^pies? most unmistakable class of such truths. kimU "A whole is equal to the sum of all its parts," A whole r , . c c equal to the is an intuitive or sen-evident truth, interred from Slim ofaii facts previously learned. For example ; having learned from experience and through the senses tmth * 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 HOW interred. 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, A11 the facts are presented are presented to the mind, and the induction to the mind. is made without the aid of other facts ; hence, 46 LOGIC. [BOOK i. AH the it is an intuitive or self-evident truth. All the axioms are . deduced i other axioms of Geometry are deduced from the same . , , r r- way. premises and by processes of inference, entirely 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 general ever self-evident, are still general propositions : so far of the inductive 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 Difference this i that, in raising the axioms of Geometry, themtnd the instances offer themselves spontaneously, and rrop^itions, without the trouble of search, and are few and qrtrediifert sim P^ e : in raising those of nature, they are in- 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 . generalized themselves spontaneously to the mind, but require from factS5 search and acuteness to discover and point out tra ^ 8 d in _ their connections, and all truths inferred from ferredfrom 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 little consequence to dis- Necessityof tinffuish the processes of reasoning by which thedistinc - J tion, being truths are inferred from facts, from those in which the basis of a 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. LOGIC. 29. Logic takes note of and decides upon Logic the sufficiency of the evidence by which truths su ^ ci e e s ncy e 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 its province. 48 LOGIC. [BOOK i. Furnishes Logic to furnish the tests by which all truths troth. that are not intuitive may be inferred from 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 data, or ultimate premises of our knowledge ; positions, nor w i t } 1 ^gjj. num ber 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 aii tests for that is, derived from facts or other truths pre- general propositions, viously known, whether 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- o ed. And since by far the greatest portion of The greatest our knowledge, whether of particular or general portion of our knowledge truths, is avowedly matter of inference, nearly the wn l e > not only of science, but of human conduct, is amenable to the authority of logic. towhat CHAP. II.] INDUCTION. 49 IN DUCTION. 30. That part of logic which infers truths from facts, is called Induction. Inductive rea soning is the application of the reasoning pro- 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 defined. the mere sum of the facts, but a conclusion drawn from them. The logic of Induction consists in classing Logic of Induction. the facts and stating the inference in 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 f^the the unknown; and any operation involving no knowntothe * unknown. 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, . a process of ization. It is that operation of the mind by ge neraiiza- which we infer that what we know to be true 4 59 LOGIC. [BOOK i. in which in a particular case or cases, will be true in all ^h^u! cases whicn resemble the former in certain as- true under s ig na bl e respects. In other words, Induction is particular circumstan- the process by which we conclude that what ccs will be . . , r , euniver- is true of certain individuals of a class is true 8aUy< of the whole class ; or that what is true at cer tain times, will be true, under similar circum stances, at all times. Induction j 32. Induction always presupposes, not only presupposes fa^ fa Q necessary observations are made with accurate and necessary the necessary accuracy, but also that the results observations. _ -11 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 serve to connec t the observations, would be to than to connect the substitute hypothesis for theory, and imasnna- observations: we must tion for proof. The connecting link must be infer from . . them. some character which really exists in the iacts 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 of the English 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 facts were HOW he his data ; from which, combining them with the general principle, that nature is steady and uni- whyhe 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 , . . difficulty 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 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 varying directly as their masses, and inversely *&****<* as tne sauares f tne i r distances, by Induction, universal jj e saw an app i e falling f rO m the tree : a mere gravitation. 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- HOW he tion. This led him to note other facts, to prose cute experiments, to observe the heavenly bodies, their until from many facts, and their connections 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 The use then brought the exact sciences to his aid, and which he made of demonstrated that this law accounted for all the science, phenomena, and harmonized the results of all ob servations. Thus, it was ascertained that the what was laws which regulate the motions of the heav- 5Ult enly bodies, as they circle the heavens, also guide the feather, as it is wafted along on the passing breeze. The ways of 33. We have already indicated the ways in ascertaining . facts are which the lacts 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 their exact nature must be ; but we and what their relations to each other, and to the inferences which flow from them ; are ques- in a11 cases when we can tions which do not admit of definite answers, draw on inference. Although no general law has yet been discov ered connecting all facts with truths, yet all the No general law. 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- Businesg mine the effects of every cause, and the causes of Induction. of all effects, if we had for all such processes general and certain laws, we could determine, what is in all cases, what causes are correctly assigned to what effects, and what effects to what causes, and we should thus be virtually acquainted with the whole course of nature. So far, then, as we How far a can trace, with certainty, the connection be- 8cience - 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. LOGIC. [BOOK i. CHAPTER III. DEDUCTION NATURE OF THE SYLLOGISM ITS USES AND APPLICATIONS. DEDUCTION. 34. WE have seen that all processes of inductive Reasoning, in which the premises are particular P re^nTng ^cts, 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 tions. Hence, a deduction is the process of Deduction reasoning by which a particular truth is inferred from other truths which are known or admitted. Deductive The formula for all deductions is found in the Syllogism, the parts, nature, and we shall now proceed to explain. Syllogism, the parts, nature, and uses of which PROPOSITIONS. Proposition, 35 - A proposition is a judgment expressed in words. Hence, a proposition is defined logi cally, " A sentence indicative :" affirming or * Section 30. CHAP. III.] PROPOSITIONS. 55 denying; therefore, it must not be ambiguous, must not be ambiguous ; for that which has more than one meaning is norimpor . in reality several propositions ; nor imperfect, j^^ 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 assume the form of a proposition. All truth and all error lie in propositions. What we call a truth, is simply a true proposition; and errors its nature, extent. 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 Embracesa11 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 nature of answers to those questions, they have actually thought they had grounds to believe. 37. The first glance at a proposition shows A proposition , . . f. , .. . , is formed by that it is formed by putting together two names. putting two Thus, in the proposition, "Gold is yellow," the names 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 born in England is denied of the man Franklin. 38. Every proposition consists of three parts : the Subject, the Predicate, and the Co- subet, P u ^ a- ^ ne su ^J ect * s tne name denoting the Predicate, person or thing of which something is affirmed and copula, 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 first, and the predicate last. The copula, in the middle, indicates the act oi judgment, and is the sign denoting that there is ar affirmation or denial. Thus, in the proposi- subject ti n, " The earth is round ;" the subject is the defined. 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 Predicate, pb* se is) predicated : the word is, which serves as connecting mark between the subject and th( predicate, to show that one of them is af firmed of the other, is called the Copula. The mustTe cc ^ a must ke either is, or is NOT, the substan- is or is NOT. th verb being the only verb recognised by AH verbs Logic. All other verbs are -resolvable, by means resolvable r ,1 i , ,, , into "to be." ol tne verb to be > an d a participle or adjective. For example : " The Romans conquered :" CHAP. III.] SYLLOGISM. 57 the word " conquered" is both copula and predi- Examples cate, being equivalent to "were victorious." Co ula 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- Aproposition is either course, in which something is affirmed or denied affirmative c , . 17 . . i T T i or negative 01 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." \ 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, ,, , , ,, , . , . , the copula is, consists of the woras "is NOT, which is he ISNOT sign of negation ; " is" being the sign of affi .1- ation. SYLLOGISM. 40. A syllogism is a form of stating the c .1- A syllogism consists of nection which may exist, for the purpose of three propo- reasoning, between three propositions. Hence, to a legitimate syllogism, it is essential that J Two sire there should be three, and only three, proposi- admitted; 58 LOGIC. [BOOK i. and the third tions. Of these, two are admitted to be true, fromuiem. anc ^ are called the premises : the third is proved from these two, and is called the conclusion. For example : " 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. Major Term Of the two terms of the conclusion, the Predi- defined. cate (d etesta bi e ) i s ca li e d t h e major term, and the Subject (Caesar) the minor term ; and these two terms, together with the term "tyrant," make up the three propositions of the syllogism, Minor Term. eacn term being used twice. Hence, every syllogism has three, and only three, different terms. premL ^^ e premiss, into which the Predicate of the defined, conclusion enters, is called the major premiss ; Minor the other is called the minor premiss, and con- Premiss. other term, common to the two premises, and with which both the terms of the conclusion were separately compared, before they were compared MiddleTerm. 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, Mil j or premiss, Minor premiss, and Middle Term. " All tyrants are detestable," is the major premiss, and " Caesar was a tyrant," the minor premiss, and " tyrant" the middle term. 41. The syllogism, therefore, is a mere for mula for ascertaining what may, or what may a mere 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 HOW applied. Major Premiss), and the given subject of the second (a Minor Premiss), 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 seen that the Major Premiss always declares, premiss. in a general way, such a relation between the Major Term and the Middle Term ; and the Mi- or the Minor, nor Premiss declares, in a more particular way, such a relation between the Minor Term and the Middle Term, as that, in the Conclusion, or the Middle Term. the Minor Term must be put under the Major Term ; or in other words, that the Major Term must be predicated of the Minor Term. 60 LOGIC. [BOOK i. ANALYTICAL OUTLINE OF DEDUCTION. Reasoning 42. In every instance in which we reason, aed 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. rectly conducted), whether we use the inductive process or the deductive formulas. Of course it cannot be supposed that every Everyone one is even conscious of this process in his own not conscious , , , ofthe mind; much less, is competent to explain the process, principles on w hich it proceeds. This indeed is, The same for and cannot but be, the case with every other every other process. Process 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 Elements and before a system of Grammar could be devised ; and musi cal compositions, previous to the sci- ^ f ^^ ThlS b 7 the Way, Serves tO 6X- tiou and pose the futility of the popular objection against classification of principles. -Logic ; viz. that men may reason very well who know nothing of it. The parallel instances ad duced show that such an objection may be urged CHAP. III.] ANALYTICAL OUTLINE. 61 in many other cases, where its absurdity would Logic 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 sameness of the reasoning attainment of a just view of the nature and ob- process ject of Logic, is the not fully understanding, or 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, AH kinds of and metaphysical, and political, &c., were essen- re< ^ l e n f n are tially different from each other, that is, different P riuci P le - lands 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- Reason of the prevail- dered at ; since it is evident to all, that some ing error8> 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- 62 LOGIC. [BOOK i. The reason of moved, by considering the parallel instance of Arithmetic ; in which every one is aware that by example, fa G process of a calculation is not affected bv which shows > that the rea- the nature of the objects whose numbers are soning process is before us ; but that, for example, the multipli- always the . c -. . same. cation oi 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 logic V a S T simple s y stem of L g ic ma 7 b e applied to all peculiar subjects whatever, are yet disposed to view it method of reasoning: as a peculiar method of 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 any one were to mistake grammar for a pecu- correctly : i 7 liar 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 reasoning; the logician s object being, not it lays down rules, not to lay down principles by which one may reason, which may, but which but by which all must reason, even though they mus tbe 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 ,, reasoning ceived that the operation of reasoning is in all Sh0 uidbe cases the same, the analysis of that operation *"*!** 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 .... . analysis is or design; and since even those who are not necessaryto misled by these fallacies, are so often at a loss furuish 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 i. rules for the eral rules of reasoning, applicable to all cases; detection of , wn i cn a person might be enabled the more error and the * discovery of readily and clearly to state the grounds of his own conviction, or pf 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 wou i c i }y e analogous to those of Arithmetic, which are analogous to the rules of obviate 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 ing to be called the art of wrangling, be more the parties, in , , . . argument, to J ust ty characterized as the "art of cutting short sue< wrangling," by bringing the parties to issue at once, if not to agreement; and thus saving a waste of ingenuity: Every con- 46 j n p ursum g the supposed investigation, it will be found that in all deductive processes ever y concl usion is deduced, in reality, from two Premises, other propositions (thence called Premises) ; for thou g h one of these may be, and commonly is, su PP r essed, it must nevertheless be understood understood, as admitted ; as may easily be made evident by supposing the denial of the suppressed premiss, CHAP. III.] 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 andis had an intelligent author," though he may not be "JJj^J aware in his own mind of the existence of any ment, though one may not. other premiss, 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 validity of the argument. 47. When one of the premises is suppressed a syllogism (which for brevity s sake it usually is), the argu- with one ment is called an Enthymeme. For example : suppressed. " The world exhibits marks of design, Therefore 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 to the assertion itself, or objections to its force ment - as an argument. For example : in the above Example, instance, an atheist may be conceived either de- 5 LOGIC. [BOOK i. nvino- that the world does exhibit marks of de- Both prera- J K* must be g^^ or denying that it follows from thence that it had an intelligent author. Now it is impor- 8 und: tant to keep in mind that the only difference in the two cases is, that in the one the expressed premiss is denied, in the other the suppressed; and when for the force as an argument of either premiss both are true, ^ ^ g on fa Q o ther premiss : if both be admit- the conclu- i sion follows. te( ^ t ^e conclusion legitimately connected w T ith them cannot be denied. 48. It is evidently immaterial to the argu ment whether the conclusion be placed first or Premiss last ; but it may be proper to remark, that a toecondu- premiss placed after its conclusion is called the sion is called R eason o f j t an d [ s introduced by one of those the Reason. conjunctions which are called causal, viz. " since," "because," &c., which may indeed be employed to designate a premiss, whether it come first or niative last. The illative conjunctions " therefore," &c., llon designate the conclusion. It is a circumstance .which often occasions causes of error an j 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 Premiss and Conclusion. For Different significations example i if I say, " this ground is rich, because of the conjunctions, the trees on it are flourishing ;" or, " the trees are CHAP. III.] ANALYTICAL OUTLINE. 67 flourishing, and therefore the soil must be rich ;" Examples I employ these conjunctions to denote the con- nection of Premiss and Conclusion ; for it is are used 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 I am using the very same conjunctions to denote d ^ J.^ e the connection of cause and effect; for in this andeffect - case, the luxuriance of the trees being evident to the eye, would hardly need to be proved, but might need to be accounted for. There are, Many cases however, many cases, in which the cause is em- lnwhlc ployed to prove the existence of its effect : es pe- the reason are the same. cially in arguments relating to future events; as, for example, when from favorable weather any one argues that the crops are likely to be abun dant, the cause and the reason, in that case, co incide ; and this contributes to their being so often confounded together in other cases. 49. In an argument, such as the example i u ev ery above given, it is, as has been said, impossible for any one, who admits both premises, to avoid admit the premiss is to admitting the conclusion. But there will be fre- admit the conclusion. quently an apparent connection of premises with a conclusion which does not in reality follow GS LOGIC. [BOOK r. Apparent from them, though to the inattentive or unskilful preT^and the argument may appear to be valid ; and there conclusion ^ IQ ma nv other cases in which a doubt may exist must not be relied on. whether the argument be valid or not ; that is, whether it be possible or not to admit the prem ises and yet deny the conclusion. General rules 50. It is of the highest importance, there- f r tetk>T en " * re > to ^ a y down some re g u l ar frm to which necessary. everv 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. " All 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- detecting the c -\ i i error ceive any fallacy in such arguments, especially if enveloped in a cloud of words ; and still more, w r hen 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 spectively, with the following, the absurdity of the conclusions from which is manifest : " Every horse is an animal : A similar Sheep are not horses ; example " Therefore, they are not animals." And: " All vegetables grow ; 2d similar An animal grows ; example. Therefore, it is a vegetable." These last examples, I have said, correspond These last . \-iir correspond exactly (considered as arguments) with the for- with the mer ; the question respecting the validity of an formcr - argument being, not whether the conclusion be true, but whether it follows from the premises adduced. This mode of exposing a fallacy, by bringing forward a similar one whose conclusion is obviously absurd, is often, and very ad van- times J - resorted to. tageously, resorted to in addressing those who are ignorant of Logical rules ; but to lay down such rules, and employ them as a test, is evi- t. T n rules is the dently a sater 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 The worl( j ex hibits 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 premiss. ex hibit marks of design," that they had an intel- rn the second ligent author ; and in the other premiss, "the world" is referred to that class as comprehended what we j n ^ . now fa j s evident that whatever is said of may infer. 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, for example, conclusion. " Nothing which exhibits marks of design could have been produced by chance ; The world exhibits, &c. ; Therefore, the world could not have been produced by chance," The process the process of reasoning will be found to be the of reasoning the same, same; since 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.] ANALYTICAL OUTLINE. 71 52. On further examination, it will be found AH valid that all valid arguments whatever, which are reducible to based on admitted premises, may be easily re- 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 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- Butwhen 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 re duced into all the steps, even of the longest and most com 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 prolix form of unfolding arguments should uni- that every argument versally supersede, in argumentative discourses, should i>e the above form. 72 LOGIC. [BOOK i. thrown into the common forms of expression ; and that " to ie form of syllogism. a 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 "on ruth*** might a chemist be charged with inconsistency for making use of any of the compound sub stances that are commonly employed, without previously analyzing and resolving them into Analogy to their simple 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 substance is offered to his notice, the com- uuntiuued. position of which has not been ascertained, or in which adulteration is suspected. Now a fal- Towhata lacy may aptly be compared to some adulterated fallacy may be compared, compound ; "it consists of an ingenious mixture of truth and falsehood, so entangled, so intimate ly blended, that the falsehood is (in the chemical phrase) held in solution : one drop of sound logic low detect- is tllat test 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 every reaJ argument. principles of reasoning : the maxim resulting from 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) universally, 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 turn de omni et nullo, for the indication of which we are indebted to Aristotle, and which is the keystone of his whole logical system. It is remarkable that some, otherwise judicious wh.^^^ writers, should have been so carried away by havesaidof 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 simplicity ; simplicitya though they would probably perceive at once testof in any other 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 applicable to all the cases in question, the 74 LOGIC. [BOOK i. NO solid ob- greater is its value and scientific beauty. If, ^priTcipte 116 indeed, any principle be regarded as not thus ap- ever urged. pii ca bl e , 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 blntakra an 7 kind f proof ; but it has often been taken for granted, fa g ran t ea ; it being (as has been stated) very syllogism commonly supposed, without examination, that not a distinct kind of ar- tne syllogism is a distinct kind of argument, and ; that the rules of it accordingly do not apply, nor applicable to were intended to apply, to all reasoning what- all cases. ever, where the premises are granted or known. objection: 55. One objection against the dictum of Aris totle it may be worth while to notice briefly, for intended to tne sa k e 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 we could imagine the logi- the certainty of a cian s object to be, to increase the certainty 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 is idea should ever have occurred to one who had erroneous. even the slightest 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 phenomena must take place as they do. But it would be reckoned a mark of the grossest ignorance and . The object is stupidity not to be aware that their object is not to prove, not to prove the existence of an individual 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 general law of nature. The object of Aris- ^object of J 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 to T i i IT- i i j. which each any individual syllogism ; his design was to point case con . 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- c ailed) are in reality merely generalized facts, of erased facts, which all the phenomena coming under them are particular instances ; so, the proof drawn from 76 LOGIC. [BOOK i. The Dictum Aristotle s dictum is not a distinct demonstration a f " d j f n ^, d brought to confirm another demonstration, but is demonstra- mere iy a generalized and abstract statement of 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. FIOW to trace 56. In order to trace more distinctly the the abstract- ~ , ing and different steps of the abstracting process, by an y particular argument may be brought into the most general form, we may first take a syllogism, that is, an argument stated accurately AD argument arid at full 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 an d then somewhat generalize the expression, by ex ^ e g ^ d t by substituting (as in Algebra) arbitrary unmean- terms. i n g S y m bols 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." The reason- The reasoning, when thus stated, is no less evi- ing no less valid, 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- whatever that is affirmed of a certain entire class" ^tJTen 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 before 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- , These three tamed - members The three members into which the maxim is correspond to the three here distributed, correspond to the three propo- propositions 78 LOGIC. [BOOK i. sitions of the syllogism to which they are in tended respectively to apply. Advantage of The advantage of substituting for the terms, in a regular syllogism, arbitrary, unmeaning sym- ls, suc } 1 as 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, fa Q argument is concerned, the connection be- the essential point of the tween the premises 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 cusec i o f ma king use of these symbols for the right in using these sym- purpose of darkening his demonstrations, and bols. that too by persons not unacquainted with geom etry and algebra, is truly astonishing. syllogism 5g. It belongs, then, exclusively to a syilo- cqually true whenab- gism, properly so called (that is, a valid argu- Btract terms , , are used. m ent, so stated that its conclusiveness is evident 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, .... , the supposed unsound and sophistical, or else may be reduced argument (without any alteration of its meaning) into the li 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 when re duced to the fallaciousness becomes more evident, from its form, the fai- nonconformity to the foregoing rule. For ex- ^1^ 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- Analysis of c )) this syllogism. 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 that class; and though what is affirmed 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 r. so contained ; for it is evidently possible that it the lament ma 7 be applicable to a whole class and to some- is not good. tnm g e j ge 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 a jj wno are ca p a ble of deliberate crime are re statement implies, 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 premiss before us ; be done in . the analysis and in the analysis of an argument, we are to argument discard all consideration of what might be as serted ; contemplating only what actually is laid down in the premises. It is evident, therefore, The one that such an apparent argument as the above comply with does n t comply with the rule laid down, nor the rule. can b e so stated as to comply with it, and is consequently invalid. 60. Again, in this instance : Another " Food is necessary to life ; example. Corn is food ; Therefore corn is necessary to life :" in what the tne term "necessary to life" is affirmed of food, argument ia defective, but not universally ; for it is not said of every kind of food: the 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 cated (that is, affirmed or denied), not of the why we whole, but of a, part only of a certain class, can- not be, on that ground, predicated of whatever whatwas predicated of is contained under that class. lood - DISTRIBUTION AND NON-DISTRIBUTION OF TERMS. 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 tion 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 dtstn- . , , ,. ., *uf Mm means. food, 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- varied " * Section 15. 6 82 LOGIC. [BOOK i. thing" (not " whatever or " every thing"} " which exhibits marks of design, is the work of an intelligent author," it would not have fol- whatit l owe( i f r om the world s exhibiting marks of de- wmikt then sign, that that is. the work of an intelligent author. words mark- 62. It is to be observed also, that the words all" and " every," which mark the distribution distribution Q a te and some which marks its non- not always 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. suchpropo- "every man." Propositions thus expressed are called called by logicians " indefinite," because it is left undetermined by the form of 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 not ; consequently, every proposition, proposition ,.-,., must be whether expressed indefinitely or not, must be either understood as either "universal" or "particu- Umversal or Particular. Jar " those 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 Example of each. only a part of them is included. For example : CHAP. III.] ANALYTICAL OUTLINE. 83 "All men are sinful/ is universal: "some men This division relates to are sinful," particular ; and this division of prop- * quantity. ositions, having reference to the distribution of the subject, is, in logical language, said to be ac cording to their " quantity." 63. But the distribution or non-distribution Distribution of the predi- of the predicate is entirely independent of the quantity of the proposition ; nor are the signs " all" and " some" ever affixed to the predicate ; because its distribution depends upon, arid is Has reference 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, Whonitis distributed: undistributed. The reason of this may easily be understood, by considering that a term which The reason J 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 thus affirmed may be of much narrower extent than that other, and may therefore be far from may be a P plicable to coinciding with the whole of it. Thus it may the subject. 1-1 1 TVT aiU J Ct f 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 i. Hence, oniya all who are uncivilized are Negroes." It is ev- tt ifu^d. Went, therefore, that it is a part 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 Mtei^wuh and predicate coincide, that is, are of equal the subject: ex t en t j a s, 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 form of the angles are equilateral ;") yet this is not implied llon by the form of the expression ; 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 the predicate ,. . ,. , , , . . , is applicable predicate is applicable to the subject, it may be to the sub- affirmed, and O f course cannot be denied, of that ject, it may be affirmed subject ; and consequently, when the predicate of the sub- J ject. is denied of the subject, this implies that nt, part of that predicate is applicable to that sub ject ; that is, that the whole of the predicate is if a predicate denied of the subject : for to say, for example, is denied, of a , subject, the tn at " no beasts of prey ruminate, implies that Beasts f P rev are excluded from the whole class the suvet ^ ruminant am ma ls> a nd consequently that " no ruminant animals are beasts of prey." And CHAP. III.] ANALYTICAL OUTLINE. SL hence results the above-mentioned rule, that the Distribution distribution of the predicate is implied in nega- tive propositions, and its non-distribution in af- propositions : firmativeS. non-distribu tion in affirmatives. 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 . , . . . .,, , occur in a in a universal proposition ; since 11 that propo- un i ver sai sition be an affirmative, and the middle term be P r P sition - 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," nothing could have been proved ; since, though both these propositions are universal, the middle terms of the conclusion, term is made the predicate in each, and both are that those terms 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 O ne prem- the middle term may then be made the predicate 8(5 LOGIC. [BOOK i. uve, the mid- of that, and will thus, according to the above bemwtetbT remark, be distributed. For example : predicate of "be^dtetriiT 11 " ^ rum nant an i ma ^ s are predacious : 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, m( j ee j O f the syllogism is not that prescribed gism will not J bethatpre- by the dictum of Aristotle, but it may easily be scribed by the dictum, reduced to that form, by stating the first prop- but may be . . 1VT . . 1 reduced to it. osition thus : "IMo predacious 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. Aiiargu- 65. It is not every argument, indeed, that merits cannot . be reduced c an be reduced to this form by so short and sim- lort a pie an alteration as in the case before us. A process. * 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 mi argu- this form, without at all departing from the real may meaning and drift of it ; and the form will be CHAP. III.] ANALYTICAL OUTLINE. 87 found (though more prolix than is needed for be reduced ordinary use) the most per; argument can be exhibited. v , . i i to the pre- ordmary use) the most perspicuous in which an scribed form 66. All deductive reasoning whatever, then, AH deductive -I i i i , reasoning rests on the one simple principle laid down by res t SO nthe Aristotle, that dictura - " 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 J 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- The reason 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 writers on kept this in mind, 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, Aristotle s would have perceived that these are, in fact, its 88 LOGIC. [BOOK i. simple and highest praise i the easiest, shortest, and most evident theory, provided it answe of explanation, being ever the best. evident theory, provided it answer the purpose RULES FOR EXAMINING SYLLOGISMS. Tests of the 67. The following axioms or canons serve validity of syllogisms, as 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. Sdtest. 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- t*.a* ,-P nil test of all affirmative lidity of affirmative conclusions ; on the latter, conclusions. ;onclusions. r The second ot negative: tor, no syllogism can be faulty which does not violate 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- 1st. Every syllogism has three and only three. gismhas . three and terms > viz. the middle term and the two terms 0n tennT e of the Conclusion : the terms of the Conclusion are sometimes called extremes. Every sy.,0. 2d . Every syllogism has three and only three CHAP. III.] ANALYTICAL OUTLINE. 89 propositions; viz. the major premiss ; the minor gismhas . , three and premiss; and the conclusion. only throe 3d. If the middle term is ambiguous, there P r P sitk)Ils - Middle term are in reality two middle terms, in sense, though must not be but one in sound. ambiguou9 There are two cases of ambiguity: 1st. Where the middle term is equivocal ; that is, when used l8tcase . in different senses in the two premises. For example : " Light is contrary to darkness ; J Example. Feathers are light ; therefore, Feathers are contrary to darkness." 2d. Where the middle term is not distrib uted ; for as it is then used to stand for a part only of its significates, 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 : " White is a color ; Black is a color ; therefore, Examples. Black is white." Again : " Some animals are beasts ; Some animals are birds ; therefore, Some birds are beasts." The middle 3d. The middle term, therefore, must be dis- term must be once distrib- tributed, once, at least, in the premises ; that is, uted; LOGIC. [BOOK i. and once is by being the subject of a universal,* or predi cate of a negative ;f and once is sufficient ; since if one extreme has been compared with a part of the middle term, and another to the whole of it, they must have been compared with the same. Notermmust 4th. No term must be distributed in the con- be distribu- . 7.7 ted in the elusion which was not distributed in one of the wTiciTw Premises; for, that would be to employ the not distribu- w hole of a term in the conclusion, when you ted in a y premiss, had employed only a part of it in the premiss ; thus, in reality, to introduce a fourth term. This is called an illicit process either of the major or minor term. J For example : Example. " A ^ quadrupeds are animals, A bird is not a quadruped ; therefore, It is not an animal." Illicit process of the major. Negative 5th. From negative premises you can infer premises prove noth- 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 : Example. A fish is not a quadruped ;" " A bird is not a quadruped," proves nothing. * Section 62. f Section 63. \ Section 40. CHAP. III.] ANALYTICAL OUTLINE. 91 6th. If one premiss be negative, the conclu- ifoneprem- 7 . r . , , iss is nega- swn must be negative; lor, in that premiss the ti the middle term is pronounced to disagree with one c " lus1011 will be nega- of the extremes, and in the other premiss (which tive ; of course is affirmative by the preceding rule), to agree with the other extreme ; therefore, the extremes disagreeing with each other, the con clusion is negative. In the same manner it may andretipro 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 aix 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 beasts are not sagacious ; Some beasts are not animals.") And, for the same reason, 2dly, that if one of 2d inference, the premises be particular, the conclusion must be particular. For example : " All who fight bravely deserve reward ; -Exam le " 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 92 LOGIC. [BOOK i. - universal Premises you cannot always infer a * universal Conclusion. For example : give a uni versal con- " All gold is precious ; elusion. All gold is a mineral ; therefore, Some mineral is precious. And even when we can infer a universal, we are always at liberty to infer a particular ; since what is predicated of all may of course be pre dicated of some. F FAL LACIES. Definition of 68. By a fallacy is commonly understood a fallacy. " any unsound mode of 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 readiness ; but still we may give some hints that will lead to correct 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 gicingeneral. T , , i i Logic in 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 . 1 . , to cultivate length ; yet a familiarity with logical principles habits of tends very much (as all feel, who are really well cle ^ 90 - acquainted with them) to beget a habit of clear and sound reasoning. The truth is, in this as in many other things, there are processes going The habit fixed, we on in the mind (when we are practising any naturally foi- v . , . , . low the thing quite familiar to us), with such rapidity p rocess e 8 . 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 . follows from process of reasoning, logically stated, the con- twoantece . elusion is inferred from two antecedent propo- dent P rem - ises. sitions, called the Premises. Hence, it is man ifest, that in every argument, the fault, if there Fallacy, if any, either in be any, must be either, the promises 1st. In the premises; or, 2d. In the conclusion (when it does not follow or conclu sion, or both. from 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 assumed, and the conclusion legitimately follows from them, the fallacy is called a Material Fal- 94 LOGIC. [HOOK i. lacy, because it lies in the matter of the argu ment. when in the Where the conclusion does not follow from ion 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 arid reasoning, the fallacy is both Material and Logical. Rules for 70. In examining a train of argumentation, examining a - c r 11 i . train of ar- to ascertain if a fallacy have crept into it, the gument. following points would naturally suggest them selves : 1st Rule. 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 ? sdRuie. 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, m examining arguments for the purpose of de- CHAP. III.] 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 with the suffice to point out the general drift and purpose r ^ of the science, and to show its entire correspond ence with the reasonings in 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: does it form ters on logic have differed, whether, and in con- a part O f formity to what principles, Induction forms a part of the science ; Archbishop Whately main- taining that logic is only concerned in inferring truths from known and admitted premises, and that all reasoning, whether Inductive or Deduc tive, 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, and 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 passed upon 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 ticular use which I wished to make of logic. Leading 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, Toshowthat that mathematical reasoning conforms, in every mathemati- . ,, . cai reasoning respect, to the strictest rules oi logic, and is m- conformsto deed ^ ut ] O g[ c applied to the abstract quantities, logical rules. 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 axioms, and that these are established by induc- established. .. -T,, tive processes. The processes of 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 the quantities, facts, truths or principles, what- Two P arts f 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 Aiiinduc- be well to remark, that every induction may thrown into be thrown into the form of a syllogism, by sup- .^Jj* plying the major premiss. If this be done, we syiiogism,by admitting a shall see that something equivalent to the uni- proper major formity of the course of nature will appear as the ultimate major premiss of all inductions ; and will, therefore, stand to all inductions in the relation in which, as has been shown, the major premiss 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- HOW this : i i r r major prem- jor premiss, or any substitution tor it, is an inter- issisobtain . ence by Induction, but cannot be arrived at by ed - means of a syllogism. 7 I BOOK II. MATHEMATICAL SCIENCE, CHAPTER I. QUANTITY AND MATHEMATICAL SCIENCE DEFINED DIFFERENT KINDS OF QUAN TITY LANGUAGE OF MATHEMATICS EXPLAINED SUBJECTS CLASSIFIED UNIT OF MEASURE DEFINED MATHEMATICS A DEDUCTIVE SCIENCE. QUANTITY. 75. QUANTITY is a general term applicable Quantity defined. to every thing which can be increased or dimin ished, and measured. There are two kinds of quantity : 1st. Abstract Quantity, or quantity, the con- Abstract. ception of which does not involve the idea of matter; and, 2dly. Concrete Quantity, which embraces concrete. every thing that is material. 76. Mathematics is the science of quantity ; Mathematics that is, the science which treats of the measures of quantities and their relations to each other. It is divided into two parts : 100 MATHEMATICAL SCIENCE. [BOOK II. Pure 1st. The Pure Mathematics, embracing the Mathematics. p rincip } es o f t h e science, and all explanations of the processes by which those principles are derived from the laws of the abstract quantities, Number and Space ; and, Mixed 2d. The Mixed Mathematics, embracing the Mathematics. a ppij cat i ons o f those principles to all investiga tions and to the solution of all questions of a practical nature, whether they relate to abstract or concrete quantity. Mathematics, 77. Mathematics, in its primary significa- tto ancients: ti n > as usea< by the ancients, embraced every acquired science, and was equally applicable to all branches of knowledge. Subsequently it was restricted to those branches only which were acquired by severe study, or discipline, and its embraced aii votaries were called Disciples. Those subjects, subjects IP 1-1 which were therefore, which required patient investigation, exact reasoning, and the aid of the mathemati- ture. ca ] analysis, were called Disciplinal or Mathe matical, because of the greater evidence in the arguments, the infallible certainty of the conclu sions, and the mental training and development which such exercises produced. Pure 78. It has already been observed that the -i pure Mathematics embrace all the principles of CHAP. I.] NUMBER. 101 the science, and that these principles are de- what they duced, by processes of reasoning upon the two abstract quantities, Number and Space. All the definitions and axioms, and all the truths deduced from them, are traceable to those two sources. Here, then, two important questions TWO ques tions, present themselves : 1st. How are we to attain a clear and true Howdowe conceive of conception of these quantities ? and, the quanti ties? 2dly. How are we to represent them, and what How re re _ language are we to employ, so as to make their ^t 116111 - properties and relations subjects of investiga tion ? NUMBER. 79. NUMBERS are expressions for one or Number more things of the same kind. How do we defined, attain unto the significations of such expres- sions ? By first presenting to the mind, through tein m idea * of number. the eye, a single thing, and calling it ONE. Then presenting two things, and naming them TWO : then three things, and naming them THREE ; and so on for other numbers. Thus, we acquire primarily, in a concrete form, our elementary i t j a done by notions of number, by perception, comparison, m%lon, and reflection ; for, we must first perceive how ^ reflection. many things are numbered ; then compare what is designated by the word one, with what is Reasons. 102 MATHEMATICAL SCIENCE. [BOOK II. designated by the words two, three, &c., and then reflect on the results of such comparisons until we clearly apprehend the difference in the signification of the words. Having thus acquired, in a concrete form, our conceptions of numbers, we can consider numbers as separated from any particular objects, and thus form a conception TWO axioms of them in the abstract. We require but two the formation axioms for the formation of all numbers : ofnumbers. 1st axiom, and that the number which results will be great er by one than the number to which the one was added. 2d axiom. 2d. That one may be divided into any num ber of equal parts. Language 80. But what language are we to employ employed. as best suited to furnish instruments of thought, and the means of recording our ideas and ex- Theten pressing them to others? The ten characters, called fi g ur es, are the alphabet of this language, and the various ways in which they are com bined will be fully explained under the head Arithmetic, a chapter devoted to the considera tion of numbers, their laws and language. CHAP. I.] SPACE. 103 SPACE. 81. SPACE is indefinite extension. We ac- s P ace defined. quire our ideas of it by observing that parts of it are occupied by matter or bodies. This ena bles us to attach a definite idea to the word place. We are then able to say, intelligibly, Place: that a point is that which has place, or position a point - in space, without occupying any part of it. Hav ing conceived a second point in space, we can understand the important axiom, " A straight line is the shortest distance between two points ;" Axiom con- and this line we call length or a dimension of J^^ ime. space, 82. If we conceive a second straight line to be drawn, meeting the first, but lying in a direction directly from it, we shall have a second dimension of space, which we call breadth. If Breadth defined. these lines be prolonged, in both directions, they will include four portions of space, which make up what is called a plane surface or plane : hence, a plane has two dimensions, length and A plane breadth. If now we draw a line on either side of this plane, we shall have another dimension of space, called thickness: hence, space has three space has dimensions length, breadth, and thickness. siongt 104 MATHEMATICAL SCIENCE. [flOOK II. Figure 83. A portion of space limited by bounda- deflned. ^^ . g ca jj e( j a pig ure . If such portion of space Line defined have but one dimension, it is called a line, and may be limited by two points, one at each ex- Two kinds of tremity. There are two kinds of lines, straight MUM and anc ^ curve< ^- ^ stra ig nt l me > * s one which does curved. not c h an g e ^s direction between any two of its points, and a curved line constantly changes its direction at every point. 84. A portion of space having two dimen- faurface : 1 sions is called a surface. There are two kinds Plane, curved, of surfaces Plane Surfaces and Curved Sur faces. With the former, a straight line, having Difference. two points in common, will always coincide, however it may be placed, while with the latter Boundaries of a surface, it will not. The boundaries of surfaces are lines, straight or curved. %iid denned $ 85 ^ P or ^ on ^ s P a ce having three dimen sions, is called a solid, and solids are bounded either by plane or curved surfaces. 86. The definitions and axioms relating to space, and all the reasonings founded on them, science of make up the science of Geometry. They will all be fully treated under that head. CHAP. I.] ANALYSIS. 105 ANALYSIS. 87. ANALYSIS is a general term embracing Analysis. all the operations which can be performed on quantities when represented by letters. In this branch of mathematics, all the quantities con sidered, whether abstract or concrete, are rep- Quantities resented by letters of the alphabet, and the b let JJ operations to be performed on them are indi cated by a few arbitrary signs. The letters and signs are called Symbols, and by their com- symbols, bi nation we form the Algebraic Notation and Language. 88. Analysis, in its simplest form, takes the Anal y sis > Algebra; name of Algebra ; Analytical Geometry, the Dif- Analytical ferential and Integral Calculus, extended to in clude the Theory of Variations, are its higher and most advanced branches. 89. The term Analysis has also another sig- Term . .sis defined. nmcation. It denotes the process of separating any complex whole into the elements of which its nature, it is composed. It is opposed to Synthesis, a synthesis term which denotes the processes of first con sidering the elements separately, then combining them, and ascertaining the results of the combi nation. 106 MATHEMATICAL SCIENCE. [BOOK II. Analytical The Analytical method is best adapted to in vestigation, and the presentation of subjects in synthetical their general outlines; the Synthetical method method. . 1 is best adapted to instruction, because A exhib its all the parts of a subject separately, and in their proper order and connection. Analysis deduces all the parts from a whole : Synthesis forms a whole from the separate parts. Arithmetic, go. Arithmetic, Algebra, and Geometry are Algebra, Geometry, the elementary branches of Mathematical Sci- ence - Every truth which is established by mathematical reasoning, is developed by an arithmetical, geometrical, or analytical process, or by a combination of them. The reasoning in each branch is conducted on principles iden tically the same. Every sign, or symbol, or technical word, is accurately defined, so that to each there is attached a definite and precise idea. Thus, the language is made so exact and certain, as to admit of no ambiguity. LANGUAGE OF MATHEMATICS. 91. The language of Mathematics is mixed, mixed. Although composed mainly of symbols, which are defined with reference to the uses which are made of them, and therefore have a pre- CHAP. I.] LANGUAGE OF MATHEMATICS. 107 else signification ; it is also composed, in part, of words transferred from our common language. The symbols, although arbitrary signs, are, nev- symbols general. ertheless, entirely general, as signs and instru ments of thought ; and when the sense in which they are used is once fixed, by definition, they preserve throughout the entire analysis precise ly the same signification. The meaning of the words bor- . - . rowed from words borrowed trom our common vocabulary is Comm0 n often modified, and sometimes entirely changed, J are modilied when the words are transferred to the language and used in a technical of science. They are then used in a particular sense. sense, and are said to have a technical significa tion. 92. It is of the first importance that the Language elements of the language be clearly understood, that the signification of every w r ord or sym bol be distinctly apprehended, and that the con- , nection between the thought and the word or symbol which expresses it be so well established that the one shall immediately suggest the other. It is not possible to pursue the subtle reasonings Mathemati- f cal reason- of Mathematics, and to carry out the trams ot ^require thought to which they give rise, without entire familiarity with those means which the mind employs to aid its investigations. The child canr.otuae any language cannot read till he has learned the alphabet; 108 MATHEMATICAL SCIENCE. [BOOK II. well till we nor can the scholar feel the delicate beauties of know it. ghakspeare, or be moved by the sublimity of Milton, before studying and learning the lan guage in which their immortal thoughts are clothed. Quantities 93. All Quantities, whether abstract or con- are repre sented by crete, are, in mathematical science, presented iKUreoper- to tne mm d by arbitrary symbols. They are these? m- v ^ ewec ^ anc ^ operated on through these symbols bois. which represent them; and all operations are indicated by another class of "symbols called signs. signs. These, combined with the symbols Whatconsti wmc ^ represent the quantities, make up, as tutesthe we have stated above, the pure mathematical language. language ; and this, in connection with that which is Borrowed from our common language, forms the language of mathematical science. ^ This language is at once comprehensive and its nature, accurate. It is capable of stating the most general proposition, and presenting to the mind, in their proper order, every elementary princi- whatitac- pl e connected with its solution. By its gener- omphshes. pure and mixed sciences, and gathers into con densed forms all the conditions and relations necessary to the development of particular facts and universal truths ; and thus, the skill of the CHAP. I.] QUANTITY MEASURED. 109 analyst deduces from the same equation the ve- Extent and locity of an apple falling to the ground, and the Anli^is. verification of the law of universal gravitation. QUANTITY MEASURED. 94. Quantity has been defined, " any thing Quantity, which can be increased or diminished, and meas ured." The terms increased or diminished, are increased and easily understood, implying merely the property diminished, of being made larger or smaller. The term measured is not so easily explained, because it has only a "relative meaning. The term " measured," applied to a quantity, Measured, implies the existence of some known quantity of the same kind, which is regarded as a stand- what it ard, and with which the quantity to be meas ured is compared with respect to its extent or magnitude. To such standard, whatever it may standard: be, we give the name of unity, or unit of meas- is called unity. ure ; and the number of times which any quan tity contains its unit of measure, is the numerical value of the quantity measured. The extent or magnitude of a quantity is, therefore, merely Magnitude: ,, i r merely rela- relative, and hence, we can form no idea of it, tive . except by the aid of comparison. Space, for example, is entirely indefinite, and we measure space: parts of it by means of certain standards, called 110 MATHEMATICAL SCIENCE. [BOOK II. Measurement measures ; and after any measurement is com- ascertains re- , . lation: pletcd, we have only ascertained the relation or proportion which exists between the standard we n processor adopted and the thing measured. Hence, measure- cumparison. ment is, after all, but a mere process of comparison. weight and 95. The abstract quantities, Weight and velocity : Tr , known by Velocity, are but vague and indefinite concep- fa^ unt Q com pared with their units of meas ure, and even these are arrived at only by pro- cesses of comparison. Indeed, most of our a general .ne.uod. knowledge of all subjects is obtained in the same way. We compare together, very care fully, all the facts which form the basis of an induction; and we rely on the comparison of the terms in the major and minor premises for every conclusion by a deductive process. Quantity. 96. Quantity, as we have seen, is divided into Abstract and Concrete the abstract quan- Abstract. tity being a mere mental conception, having for its sign a number, a letter, or a geometrical concrete, figure. A concrete quantity is a physical ob ject, or a collection of such objects, and may likewise be represented by numbers, letters, or , , by the geometrical magnitudes regarded as ma- terial The number " th ^ee" is entirely abstract, expressing an idea having no connection with sented. CHAP. I.] PURE MATHEMATICS. Ill material things ; while the number " three pounds of tea," or " three apples/ presents to the mind an idea of physical objects. So, a portion of space, bounded by a surface, all the points of which are equally distant from a certain point within called the centre, is but a mental con ception of form; but regarded as a solid mass, ofthecon- ... . it gives rise to the additional idea of a material substance. PURE MATHEMATICS. 97. The Pure Mathematics are based on Pure Mathematics: definitions and intuitive truths, called axioms, which are inferred from observation and expe- what are its foundations. rience ; that is, observation and experience fur nish the information necessary to such intuitive inductions.* From these definitions and axioms, as premises, all the truths of the science are estab lished by processes of deductive reasoning ; and there is not, in the whole range of mathemat- Itste8tsof * truths: ical science any logical test of truth, but in a conformity of the conclusions to the definitions w ^ e and axioms, or to such principles as have been established from them. Hence, we see, that in what the science con- the science of Pure Mathematics, which con- gists. sists merely in inferring, by fixed rules, all the * Section 27. 112 MATHEMATICAL SCIENCE. [BOOK II. is purely truths which can be deduced from given prem- Deductive - ises, is purely a Deductive Science. The pre cision and accuracy of the definitions ; the cer tainty which is felt in the truth of the axioms ; Precision of the obvious and fixed relation between the sign e and the thing signified ; and the certain for mulas to which the reasoning processes are re duced, have given to mathematics the name of Exact . , ? Science. " Exact Science. AII reasoning 98. We have remarked that all the reason- ^lUion^" ings of mathematical science, and all the truths axioms. w hich they establish, are based on the defini tions and axioms, which correspond to the major premiss of the syllogism. If the resemblance which the minor premiss asserts to the middle Relations not term were obvious to the senses, as it is in the obvious. . . ,, proposition, " feocrates was a man, or were at once ascertainable by direct observation, or were as evident as the intuitive truth, " A whole is equal to the sum of all its parts ;" there Deductive would be no necessity for trains of reasoning, Science, and Deductive Science would not exist. Trains Trains of of reasoning are necessary only for the sake of extending the definitions and axioms to other what they cases in which we not only cannot directly ob- accomplish. serve what is to be proved, but cannot directly observe even the mark which is to prove it. CHAP. I.] PURE MATHEMATICS. 113 99. Although the syllogism is the ultimate syllogism, test in all deductive reasoning (and indeed in ^deduction. all inductive, if we admit the uniformity of the course of nature), still we 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 i v i i / - definitions, sitions established irom them, are our tests oi tests or truth: truth ; and whenever any new proposition can be brought to conform to any one of these A proposi tion : when tests, it is regarded as proved, and declared to prov ed. be true. 100. When general formulas have been When a principle framed, determining the limits within which the maybe re- deductions may be drawn (that is, what 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 are Trains of continually arising, which do not obviously come ^"^ within any formula that will settle the questions sar y- we want solved in regard to them, and it is necessary to reduce them to such formulas. This gives rise to the existence of the science They give of mathematics, requiring the highest scientific "^^ genius in those who contributed to its creation, mathematics. and calling for a most continued and vigorous 8 Hi MATHEMATICAL SCIENCE. [BOOK II. exertion of intellect, in order to appropriate it. when created. COMPARISON OF QUANTITIES. Mathematics j JQJ \y e nave seen that the pure mathe- concerned with Number matics are concerned with the two abstract and 8pa*. q Uant i t j eSj ]\ T um ber and Space. We have also Reasoning seen that reasoning necessarily involves coin- parison : hence, mathematical reasoning must consist in comparing the quantities which come from Number and Space with each other. 102. Any two quantities, compared with ties can sus tain but two each other, must necessarily sustain one of two relations : they must be equal or unequal. What axioms or formulas have we for inferring the one or the other ? AXIOMS OR FORMULAS FOR INFERRING EQUALITY. 1. Things which being applied to each other coincide, are equal to one another. Formulas 2 Things which are equal to the same thing Equality, are equal to one another. 3. A whole is equal to the sum of all its parts. 4. If equals be added to equals, the sums are equal. CHAP. I.] COMPARISON OF aUANTITIES. 115 5. If equals be taken from equals, the remain ders are equal. AXIOMS OR FORMULAS FOR INFERRING INEQUALITY. 1. A whole is greater than any of its parts. 2. If equals be added to unequals, the sums Formulas are unequal. Inequality. 3. If equals be taken from unequals, the re mainders are unequal. 103. We have thus completed a very brief outline of i i 1-1 r- **- i i Mathematics and general analytical view of Mathematical completed. Science. We have endeavored to point out the character of the definitions, and the sources as well as the nature of the elementary and in tuitive propositions on which the science rests ; What fea tures have the kind of reasoning employed in its creation, been and its divisions resulting from the use of dif ferent symbols and differences of language. We shall now proceed to treat the subjects separ ately. rilAP. II.] ARITHMETIC FIRST NOTIONS. 117 CHAPTER II. ARITHMETIC SCIENCE AND ART OF NUMBERS. SECTION I. INTEGER UNITS. FIRST NOTIONS OF NUMBERS. 104. THERE is but a single elementary idea But one eie- - ,, ... mentary idea in the science of numbers: it is the idea of the in numbers. UNIT ONE. There is but one way of impressing HOW im- this idea on the mind. It is by presenting to the senses a single object ; as, one apple, one peach, one pear, &c. 105. There are three signs by means of Three signs which the idea of one is expressed and commu- i ng it. nicated. They are, 1st. The word ONE. Aword - 2d. The Roman character I. Roman character: 3d. The figure 1. Flgure _ 118 MATHEMATICAL SCIENCE. [BOOK II. New ideas 106. If one be added to one, the idea thus by adding arising is different from the idea of one, and is one 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. The ex- Theexpres- pressions for these, and similar ideas, are called sions are numbers, numbers i hence, Numbers NUMBERS are expressions for one or more defined. things of the same kind. IDEAS OF NUMBERS GENERALIZED. ideas of 107. If we begin with the idea of the nurn- numbers , generalized, tor one, and then add it to one, making two ; and then add it to two, making three ; and then to three, making four ; and then to four, making HOW formed, five, and so on ; it is plain that we shall form a series of numbers, each of which will be greater unity the by one than that which precedes it. Now, one nwe 8 r unit y j is t * le basis of this series of numbers, 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 ; y- 2d. By the Roman characters; and, 3d way. 3d. By figure s. CHAP. II.] ARITHMETIC UNITY. 119 108. Since all numbers, whether integer or AH numbers fractional, must come from, and hence be con- one: 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 . one idea that regarded as made up 01 units (and all numbers i 9p ureiyeie- 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 notions are complex. 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 complex. 109. 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 -,.... . number may analyze the units of which it is made up, it may be regarded th en be regarded as a simple or incomplex idea; as " 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. 110. When we name a number, as twenty what is ne- feet, two things are necessary to its clear appre- "^j^j of a number 120 MATHEMATICAL SCIENCE. [BOOK II. First. 1st. A distinct apprehension of the single thing which forms the basis of the number ; and, second. 2d. A distinct apprehension of the number of times which that thing is taken. The basis of The single thing, which forms the basis of the the uumber i n i T A n i number, is called UNITY, or a UNIT. It is called IS UNITY. when H is unity, when it is regarded as the primary basis called UNITY, Q f the num ber ; that is, when it is the final stand ard to which all the numbers that come from it and when a are referred. It is called a unit when it is re garded as one of the collection of several equal things which form a number. Thus, in the ex ample, one foot, regarded as a standard and the basis 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. OF SIMPLE AND DENOMINATE NUMBERS. Abstract 111. A simple or abstract unit, is ONE, with out regard to the kind of thing to which the term one may be applied. Denominate A denominate or concrete unit, is one thing named or denominated ; as, one apple, one peach, one pear, one horse, &c. * U2 Number as such > has no reference to the particular things numbered. * But to dis- CHAP. II.] ARITHMETIC ALPHABET. 121 tinguish numbers which are applied to particular to the things units from those which are purely abstract, we call the latter Abstract or Simple Numbers, simple and 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. 113. 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 words, 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- Grammar lished connection between words as the signs of ideas. ARITHMETICAL ALPHABET. 114. The arithmetical alphabet consists of Arithmetical Alphabet. ten characters, called figures. They are, Naught, One, Two, Three, Four, Five, Six, Seven, Eight, Nine, 0123456789 122 MATHEMATICAL SCIENCE. [fiOOK II. and each may be regarded as a word, since it stands for a distinct idea. WORDS SPELLING AND READING IN ADDITION. one cannot 115. The idea of one, being elementary, the character 1 which represents it, is also element ary, and hence, cannot be spelled by the other characters of the Arithmetical Alphabet ( 114). But the idea which is expressed by 2 comes from spelling by the addition of 1 and 1 : hence, the word repre- the arithmetical sented by the character 2, may be spelled by 1 and 1. Thus, 1 and 1 are 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. :am p ies. Four ^ s gp^fe^ j and 3 ^ ^ . 3 and I 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. AH numbers 116. In a similar manner, any number in may be . , J speiied in a arithmetic may be spelled; and hence we see vay - that the process of spelling in addition consists simply, in naming any two elements which will make up the number. All the numbers in ad- CHAP. ii. ] ARITHMETIC READINGS. 123 dition are therefore spelled with The reading consists in naming two syllables, only the word Reading: in which expresses the final idea. Thus, what it con sists. 1 2 3 4 5 6 7 8 9 Examples. 1 1 1 1 1 1 1 1 1 1 One two three four five six seven eight 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 four, &c. 1 2 3 4 5 6 7 8 9 10 Three, four, 2 2 2 2 2 2 2 2 2 2 *. 1 2 3 4 5 6 7 8 9 10 Four, five, 3 3 3 3 3 3 3 3 3 3 &c. 1 2 3 4 5 6 7 8 9 10 Five, six, &c. Six, seven, 1 2 3 4 5 6 7 8 9 10 5 5 5 5 5 5 5 5 5 5 &c. 1 2 3 4 5 6 7 8 9 10 Seven, eight, 6 6 6 6 6 6 6 6 6 6 &c. 1 2 3 4 5 6 7 8 9 10 Eight, nine, 7 7 7 7 &c. 1 2 3 4 5 6 7 8 9 10 Nine, ten, &.c. 8 8 8 8 8 8 8 8 8 8 124 MATHEMATICAL SCIENCE. [l3OOK II. Ten, eleven, &C. Eleven, twelve, &c. Example for reading in Addition. 123456789 10 9999999999 123456789 10 10 10 10 10 10 10 10 10 10 10 117. In this example, beginning 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 : 878 421 679 354 764 setting down the 9 and carrying the 2, we say, 9, 12, 18, 22, 30: 3096 and setting down the 30, we have the entire sum AII examples 3096. All the examples in addition may be done so solved. in a similar manner. Advantages of reading. 1st. stated. 2d. stated. 118. The advantages of this method of read ing over spelling are very great. 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 CHAP. II.] ARITHMETIC WORDS. 125 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. stated. 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. 119. It must not be supposed that the read- Reading comes after ing can be accomplished until the spelling has spelling. first been learned. In our common language, we first learn the same as in our common alphabet, then we pronounce each letter in a language. word, and finally, we pronounce the word. We should do the same in the arithmetical reading. WORDS SPELLING AND READING IN SUBTRACTION. 120. The processes of spelling and reading same princi ple applied which we have explained in the addition of m subtree- numbers, may, with slight modifications, be ap plied in subtraction. Thus, if we are to subtract 126 MATHEMATICAL SCIENCE. [COOK ii. 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 once. Hence, the reading, in subtrac- Subtraction . 7 ,-1 j -i 7 explained. tlon > ls simply naming the word, which expresses the difference between the subtrahend and min uend. Thus, we may read each word of the following one hundred combinations. READINGS. One from 1 2 3 4 5 6 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 2 2 Three from 3 4 5 6 7 8 9 10 11 12 three, &c. 3 3 3 3 3 3 3 3 3 3 Four from 4 5 6 7 8 9 10 11 12 13 four, &c. 4 4 4 4 4 4 4 4 4 4 Five from 5 6 7 8 9 10 11 12 13 14 five, &c. 5 5 5 5 5 5 5 5 5 5 i Six from six, 6 7 8 9 10 11 12 13 14 15 &c. 6 6 6 6 6 6 6 6 6 6 Seven from 7 8 9 10 11 12 13 14 15 16 eeven, &c. 7 7 7 7 CHAP. II.] ARITHMETIC SPELLING. 127 8 9 10 11 12 13 14 15 16 17 Eight from 8888888888 eight, &c. 9 10 11 12 13 14 15 16 17 18 Ninefrom 9999999999 nine &c - 10 11 12 13 14 15 16 17 18 19 Ten from ten, 10 10 10 10 10 10 10 10 10 10 &c * 121. It should be remarked, that in subtrac tion, as well as in addition, the spelling of the spelling pre- . cedes reading words must necessarily precede their reading. m subtrac- 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. 122. Spelling in multiplication is similar to spelling in . Multiplica- the corresponding process in addition or subtrac- uon. 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 2 are 4 ; two times 3 are 6 ; two times 4 are 8 ; two times 5 are 10; two times 6 are 12; two 128 MATHEMATICAL SCIENCE. [fiOOK II. times 7 are 14; two times 8 are 16; 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. Onceoneis 12 11 10 987654321 Twotimesi 12 1 1 10 9 8 7 6 5 4 3 2 1 are 2, &c. Threetimesl 12 1 1 10 9 8 7 6 5 4 3 2 1 are 3, &c. o Fourtimesl 12 1 1 10 9 8 7 6 5 4 3 2 1 are 4, &c. A Fivetimesl 12 11 10 9 8 7 6 5 4 3 2 1 are 5, &c. 12 11 10 9 8 7 6 5 4 3 2 1 are six, &c. r* Seven timea 121110987654321 1 are 7, &c. 1 12 11 10 987654321 are8, &c . CHAP. II.] ARITHMETIC READING. 12 11 10 9 8 7 6 5 4 3 2 1 Nine timos 1 are 9, &c. 12 11 10 987654321 Ten times I JQ are 10, &c. 12 11 10 987654321 Eleven times 1 -i 1 are 11, <fcc. 12 11 10 987654321 Twelve times lareJ2,&c. SPELLING AND READING IN DIVISION. 123. In all the cases of short division, the in short Dh-i- . ft s n we ma y 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 6 8 10 12 16 18 22 two three four five six eight nine eleven. In a similar manner, all the words, expressing i na ii cases, the results in short division, may be read. READINGS. 2)2 4 6 8 10 12 14 16 18 20 22 24 Twoins, once, &c. 3)3 6 9 12 15 18 21 24 27 30 33 36 Three in 3, once, &c. 4)4 8 12 16 20 24 28 32 36 40 44 48 Fourin4, once, fcc. 130 MATHEMATICAL SCIENCE. [BOOK II. Five in 5, 5)5 10 15 20 25 30 35 40 45 50 55 60 once, &.c. Fix in 6, 6)6 12 18 24 30 36 42 48 54 60 66 72 once, &c. Feven 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)lQ 20 30 40 50 60 70 80 90 100 110 120 once, &c. Eleven in ii, 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. .- r- TJNITS INCREASING BY THE SCALE OF TENS. The idea of a 124. The idea of a particular number is ne- particular number is ccssanly complex ; for, the mind naturally asks : 1st. What is the unit or basis of the number? and, 2d. How many times is the unit or basis taken ? What a fig- 125. A figure indicates how many times a ure indicates. unit is taken. Each of the ten figures, however written, or however placed, always expresses as many units as its name imports, and no more ; nor does the j%wre itself at all indicate the kind CHAP. II.] ARITHMETIC SCALE OF TENS. ] 131 of unit. Still, every number expressed by one or more figures, has for its basis either the abstract 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 place where the figure is written. The number of units which may be expressed by either of the ten figures, is indicated by the name of the figure. If the figure stands alone, and the unit is not denominated, the basis of the number is the abstract unit 1. 126. If we write on the right of j L *^ 1, we have ) 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 of the second order. Again ; if we write two O s on the ) Number has one for its basis. Number ex pressed by a single figure. How ten ia written. Unit of the second order. How to write one hundred. A unit of the third order. Laws when figures are written by the side of each other. 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 hundred times as great as the unit 1. 127. If three 1 s are written by j Sill, V v , * Section 111. 132 MATHEMATICAL SCIENCE. [fiOOK II. the ideas, expressed in our common language, 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. Third. 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 eacn other, the arithmetical language establishes a relation between the units of their places : that ^ 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 O s as a scale Numeration. , thus : a 3 s rf a f 3 3 ra 3 .q -S ,0 j The units of 000, 000, 000, 000 place deter- the unit of each place is determined, as well CHAP. II.] ARITHMETIC SCALE OF TENS. 133 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 01 units, ot 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. 128. It should be observed, that a figure A figure has being a character which represents value, can itsein 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 unit ~ , . , , . A , c i isdeter- 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 long Figure, has . ,, . no local in use, is, therefore, without signification when value> applied to a figure : the term " local value," being applicable to the unit of the place, and Term app ii- _ 1-1 i i cable to not to the figure which occupies the place. unit ofplace , 129. Federal Money affords an example of a Federal Money. * Section 199. 134 MATHEMATICAL SCIENCE. [BOOK II. itsdenomina- series of denominate units, increasing according to the scale of tens : thus, 4 c of +? be a g g id ca o .a a P3 ft fi U 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 and 1 mill. Thus, we may read the number with either of 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 130. Examples in reading figures : ist. Example. If we have the figures - - - - 89 we may read them by their smallest unit, and say eighty-nine ; or, we may say 8 tens and 9 units. ad. 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. 135 may be read, seventy-four thousand eight hun- various read ings 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 u the number T TI -. read by its the lowest unit of that period, r or example, in lowest im[L 874,967,847,047, Example. we read 874 billions 967 millions 847 thousands and 47. UNITS INCREASING ACCORDING TO VARYING SCALES. 131. If we write the well-known signs of Methods of writing fig- the English money, and place 1 under each de- uri , s havm ., i 11 i " different nomination, we shall have denominate units. . *. d. f. 1111 Now, the signs . s. d. and/, fix the value of HOW the value of each the unit 1 in each denomination; and they also unit is fixed. 136 MATHEMATICAL SCIENCE. [BOOK IJ. what the determine the relations which subsist between liinimnge expresses. the different units. For example, this simple language expresses these ideas : The units of 1st. That the unit of the right-hand place is the places. ^ f art hing 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 make one unit of the next higher; 12 of the second, one of the third ; and 20 of the third, increase. one of the fourth. The units m If we take the denominate numbers of the Avoirdupois . . , . . , weight. Avoirdupois weight, we have Ton. cwt. qr. Ib. oz. dr. 111111; changes in in which the units increase in the following the value of , , the units. man ner : viz. the second unit, counting from 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, uowthe scale The scale, therefore, for this class of denominate varies. numbers varies according to the above laws. A different If we take any other class of denominate scale for each system, 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 INTEGER UNITS. 137 But in all the formations, we shall recognise The method the application of the same general principles. the c . i} ^ the There are, therefore, two general methods of ^jj^J 111 forming the different systems of integer num- i r - mi e Two systems oers irom the unit one. Ihe nrst consists m O f forming preserving a constant law of relation between ln 1^ 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 B ys- of known, though varying laws of change in the unities. These changes in the unities produce change in the the entire system of denominate numbers, each fj^inj the class of which has its appropriate scale, and the changes among the units of the same class are indicated by the different degrees of its scale. INTEGER UNITS OF ARITHMETIC. 132. There are four principal classes of units Four classes ... of units. in arithmetic : 1st. Abstract, or simple units ; 1st. class. 2d. Units of Currency ; 2d. class. 3d. Units of Weight; and 3d. class. 4th. Units of Measure. 4th . clasSi First among the Units of arithmetic stands the simple or abstract unit 1. This is the basis Abstract unit of all simple numbers, and becomes the basis, OI 138 MATHEMATICAL SCIENCE. [BOOK II. Thebasi 9 of also, of all denominate numbers, by merely na- denominate numbers ; "" ming, in succession, the particular things to which it is applied. AISO, the ba- It is also the basis of all fractions. Merely as the unit 1, it is a whole which may be divided according to any law, forming every variety of inate - 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. 133. It has been remarked* that we can Must appre- f orm no distinct apprehension of a number, un bend 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 fuiiy explain- learners, a clear and distinct idea of the actual value of the unit of every number with which they have to do. If it be a number expressing \_ currency, one or more of the coins should be pressing cur- exhibited, and the value dwelt upon: after which, rency. distinct notions of the other units can be ac quired by comparison. Exhibit the ^ ^e num ber be one of weight, some unit 11 be should be exhibited, as one pound, or one ounce, and an idea of its weight acquired by actually * Section 110. CHAP. II.] ARITHMETIC - FEDERAL MONEY. 139 lifting it. This is the only way in which we can learn the true signification of the terms. If the number be one of measure, either And also, if it linear, superficial, liquid, or solid, its unit should meas ure. also be exhibited, and the signification of the term expressing it, learned in the only way in which it can be learned, through the senses, and by the aid of a sensible object. FEDERAL MONEY. 134. The currency of the United States is currency of called Federal Money. Its units are all denomi nate, being 1 mill, 1 cent, 1 dime, 1 dollar, ] eagle. The law of change, in passing from one Law of change in the unit to another, is according to the scale of tens, Hence, this system of numbers may be treated, J J How these in all respects, as simple numbers ; and indeed numbers 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 geu- . , r erallyread. cents, and mills a comma 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 decimals of the mill. The number may, how- 140 MATHEMATICAL SCIENCE. [BOOK II. The number ever, be read in any other unit ; as, 864849 read in various ways. mills ; or, 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- 135. The units of English, or Sterling Mo ney, are 1 farthing, 1 penny, 1 shilling, and 1 pound. scale of the The scale of this class of numbers is a varying scale. Its degrees, in passing from the unit of the lowest denomination to the highest, are four, HOW it twelve, and twenty. For, four farthings make changes. , .. one penny, twelve pence one shilling, and twenty shillings one pound. AVOIRDUPOIS WEIGHT. units in 136. The units of the Avoirdupois Weight Avoirdupois. 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 degrees, in passing from the unit of the lowest denomination to the highest, are sixteen, sixteen, twenty-five, four, and twenty. Variation in For, sixteen drams make one ounce, sixteen >ee? - ounces one pound, twenty-five pounds one quar- CHAP. II.] ARITHMETIC UNITS OF LENGTH. 141 ter, four quarters one hundred, and twenty hun dreds one ton. TROY WEIGHT. 137. The units of the Troy Weight are, 1 Unit9 in Troy Weight. grain, 1 pennyweight, 1 ounce, and 1 pound. The scale is a varying scale, and its degrees, scale: in passing from the unit of the lowest denomina- Its <\ eSTec9f tion to the highest, are twenty-four, twenty, and twelve. APOTHECARIES WEIGHT. 138. The units of this weight are, 1 grain, 1 units in Apothecaries scruple, 1 dram, 1 ounce, and 1 pound. weight. The scale is a varying scale. Its degrees, in Scalo: passing from the unit of the lowest denomina- it* degrees tion to the highest, are twenty, three, eight, and twelve. UNITS OF MEASURE. 139. There are three units of measure, each Three units of measure. differing in kind from the other two. They are, Units of Length, Units of Surface, and Units of Solidity. UN ITS OF LENGTH. 140. The unit of length is used for measur- units of length. ing lines, either straight or curved. It is a 142 MATHEMATICAL SCIENCE. [BOOK II. The stand- straight line of a given length, and is often called ard. the standard of the measurement. what units The units of length, generally used as stand- are taken. ards, are 1 inch, 1 foot, 1 yard, 1 rod, 1 furlong, and 1 mile. The number of times which the idea of unit, used as a standard, is taken, considered in length. connection with its value, gives the idea of the length of the line measured. Units of surface. What the unit of surface is. 1 square foot. Its connection with the unit of length. Square feet in a square yard. UNITS OF SURFACE. 141. 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 o 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.] ARITHMETIC DUODECIMAL UNITS. 143 contains the square foot nine times. The square square rod rod or square mile may also be used as the unit square mile, of surface. The number of times which a surface contains Area or . . , contents of a its unit of measure, is its area or contents ; and 8U rface. this number, taken in connection with the value of the unit, gives the idea of its extent. Besides the units of surface already considered, there is another kind, called, DUODECIMAL UNITS. 142. The duodecimal units are generally Duodecimal used in board measure, though they may be used in all superficial measurements, and also in solid. The square foot is the basis of this class of Their basis. units, and the others are deduced from it, by a descending scale of twelve. 143. It is proved in Geometry, that if the what number of linear units in the base of a rectan- gle be multiplied by the number of linear units in the height, 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- Ho witis ex- ing, that "feet multiplied by feet give square pre feet," and "yards multiplied by yards give square 144 MATHEMATICAL SCIENCE. [BOOK II. yards." But as feet cannot be taken feet times, nor yards yard times, this language, rightly un derstood, is but a concise form of expression for the principle stated above. conclusion. With this understanding of the language, we say, that 1 foot in length multiplied by 1 foot in height, gives a square foot ; and 4 feet in length multiplied by 3 feet in height, gives 12 square feet. Examples in 144. If nOW, 1 foot ill the muIlipH- cationoffeet length be multiplied by 1 inch by feet and , inches. = T5 f a foot in height, the product will be one-twelfth of a square foot ; that is, one- twelfth of the first unit: if it be multiplied by 3 inches, the product will be Generaiiza- three-twelfths of a square foot; and similarly for a multiplier of any number of inches. inches by If, now, we multiply 1 inch by 1 inch, the inches. product may be represented by 1 square inch : iiowtheunits that is, by one-twelfth of the last unit. Hence, change, and what they tke units of this measure decrease according to the scale of 12. The units are, First. 1st. Square feet arising from multiplying feet by feet. second. 2d. Twelfths of square feet arising from mul tiplying feet by inches. CHAP. II.] ARITHMETIC - UNITS 115 3d. Twelfths of twelfths arising from multi- Third. plying inches by inches. The same remarks apply to the smaller di- visions of the foot, according to the scale of twelve. The difficulty of computing in this measure Difficulty. arises from the changes in the units. UNITS OF SOLIDITY. 145. It has already been stated, that if length be multiplied by breadth, the product may be represented by units of surface. It is ~ Tiii also proved, in Geometry, that if the length, breadth, and height of any regular solid body, of a square form, be multiplied together, the product may be represented by solid units whose number is equal to this product. Each solid unit is a cube constructed on the linear unit as an edge. Thus, if the linear unit be 1 foot, the solid unit will be 1 cubic or solid foot ; that is, a cube constructed on 1 foot as an edge ; and if it be 1 yard, the unit will be 1 solid yard. The three units, viz. the unit of length, the unit of surface, and the unit of solidity, are es- sentially different in kind. The first is a line of a known length ; the second, a square of a known side : and the third, a solid, called a 10 units of sou- what is proved in Go- O metry in re- g solid units. Examples. The three t iaiiy duier- what they J 116 MATHEMATICAL SCIENCE. [BOOK II. Generally cube, of a known base and height. These are used the units used in all kinds of measurement- Duodecimal excepting only the duodecimal system, which has already been explained. LIQUID MEASURE. units of LI- 146. The units of Liquid Measure are, 1 quid Meas- . q uar t, 1 gallon, 1 barrel, 1 hogs head, 1 pipe, 1 tun. The scale is a varying scale. Its degrees, in passing from the unit of owitva- the lowest denomination, are, four, two, four, thirty-one and a half, sixty-three, two, and two. DRY MEASURE. Units of Dry 147. The units of this measure are, 1 pint, 1 quart, 1 peck, 1 bushel, and 1 chaldron. The Decrees of degrees of the scale, in passing from units of the the scale. . . lowest denomination, are two, eight, lour, and thirty-six. TIME. Units of 148. The units of Time are, 1 second, 1 Time. minute, 1 hour, 1 day, 1 week, 1 month, 1 year, Degrees of and 1 century. The degrees of the scale, in the scale. passing from units of the lowest denomination to the highest, are sixty, sixty, twenty-four, seven, four, twelve, and one hundred. CHAP. II.] ARITHMETIC ADVANTAGES. 147 CIRCULAR MEASURE. 149. The units of this measure are, 1 sec- units of cir cular Meas- ond, 1 minute, 1 degree, 1 sign, 1 circle. The ure. degrees of the scale, in passing from units of the Degrees of the Scale. lowest denomination to those of the higher, are sixty, sixty, thirty, and twelve. ADVANTAGES OF THE SYSTEM OF UNITIES. 150. It may well be asked, if the method Advantages of the system. 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 1st. Teaches analysis of all numbers, and unfolds to the mind O f 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 2d. Points out a definite and distinct idea of every number, by relation.- pointing out the relation between it and the unit from which it was derived. 3d. By presenting constantly to the mind the 3d. constant- idea of the unit one, as the basis of all numbers, the idea of the mind is insensibly led to compare this unit muty> with all the numbers which flow from it, and 148 MATHEMATICAL SCIENCE [BOOK II. then it can the more easily compare those num bers with each other. 4th. Explains 4th. It affords a more satisfactory analysis, and a better understanding of the four ground ru i es> a nd indeed of all the operations of arith metic, than any other method of presenting the subject. ground rules. FOUR GROUND RULES. system Examples. 151. Let us take the two following examples m Addition, the one in simple and the other in denominate numbers, and then analyze the pro cess of finding the sum in each. SIMPLE NUMBERS. "874198 36984 3641 914823 DENOMINATE NUMBERS cwt. qr. lb. oz. dr. 3 3 24 15 14 6 3 23 14 8 10 3 23 14 6 Process of But ouo In both examples we begin by adding the units f t ^ ie l west denomination, and then, we divide their sum by so many as make 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 principle being precisely the same in both examples. We see, in these examples, an CHAP. II.] ARITHMETIC SUBTRACTION. 149 illustration of a general principle of addition, units of the viz. that units of the same kind are always added unite . together. 152. Let us take two similar examples in system -, . applied in Subtraction. subtraction. SIMPLE NUMBERS. DENOMINATE NUMBERS. 8403 , A far. 3298 6972 5105 3 10 8 4 2 18 10 2 In both examples we begin with the units of The method of performing the lowest denomination, and as the number in the examples. the subtrahend is greater than in the place di rectly above, we suppose so many to bte added in the minuend as make one unit of the 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 , _ same for pies. Also, that units ol any denomination in aii examples. the subtrahend are taken from those of the same denomination in the minuend. 153. Let us now take similar examples in !! Multiplication. 150 MATHEMATICAL SCIENCE. [BOOK II. SIMPLE NUMBERS. DENOMINATE NUMBERS. Examples. 87464 ib 3 9 gr. 5 9 7 6 2 15 437320 5 48 3 2 1 15 Method of In these examples we see, that we multiply, in performing success i orlj e ach order of units in the multipli- the examples. can d by the multiplier, 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 for all examples, same in both examples. 154. Finally, let us take two similar exam ples in Division. SIMPLE NUMBERS. DENOMINATE NUMBERS. 3)874911 s . d far , 291637 3)8 4 2 1 2 14 8 3 Principles in- We begin, in both examples, by dividing the voived: un i ts o f t he 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 next lower denomination. We then add in that de- The same as in the nomination, and continue the division through other rules. n i i i ail the denominations to the last the principle being precisely the same in both examples. CHAP. II.] ARITHMETIC FRACTIONS. 151 SECTION II. FRACTIONAL UNITS. FRACTIONAL UNITS. SCALE OF TENS. 155. IF the unit 1 be divided into ten equal Fraction one- parts, each part is called one tenth. If one of Jjj"^. these tenths be divided into ten equal parts, each part is called one hundredth. If one of the hundredth ; hundredths be divided into ten equal parts, each One part is called one thousandth ; and corresponding thousandth - names are given to similar parts, how far soever Generaiiza- the divisions may be carried. Now, although the tenths which arise from Fractions are dividing the unit 1, are but equal parts of 1, whole things. 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. MATHEMATICAL SCIENCE. [BOOK II. Fractional 156. Adopting a similar language to that units of the . ill i r first order; used in integer numbers, we call the tenths, irac- units of the tional units of the f irst order > the hundredths, fractional units of the second order ; the thou sandths, fractional units of the third order ; and so on for the subsequent divisions. Language for Is tnere an 7 arithmetical language by which fractional t h ese fractional units may be expressed ? The units. J decimal point, which is merely a dot, or period, whatitfixes. indicates the division of the unit 1, according to the scale of tens. By the arithmetical language, Names of the the unit 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. sc 316 - The scale for decimals, therefore, is .000000000, &c. ; in which the unit of each place is known as soon as we have learned the signification of the language. If, therefore, we wish to express any of the parts into which the unit 1 may be divided, ac- Any decimal cordin g to the scale of tens, we have simply to LTx^reS Select from the al P habet > the figure that will oy this scale. express the number of parts, and then write it in CHAP. II.] ARITHMETIC FRACTIONS. 153 the place corresponding to the order of the unit, where any Thus, to express four tenths, three thousandths, written. eight ten-thousandths, and six millionths, we write .403806 ; Example. and similarly, for any decimal which can be named. 157. 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 crease from to the same scale, from right to left. This is the rig ht to left. same law of increase as that which connects the units of place in simple numbers. Hence, simple consequence. numbers and decimals being 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 ; since ten units, in the place of tenths, make the unit one in the place next to the left. Example. FRACTIONAL UNITS IN GENERAL. 158. If the unit 1 be divided into two equal Ahai parts, each part is called a half. If it be divided A fourth. A fifth. Generally. These units are whole things. Examples. MATHEMATICAL SCIENCE. [BOOK II. 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. 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 tional units. Each is an exact part of the unit 1, and has a fixed relation to it. 159. Is there any arithmetical language by which these fractional units can be expressed ? Language for The bar, written at the right, is the fractions. sign which denotes the division of the unit 1 into any number of equal parts. TO express If we wish to express the number of equal 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. Have a rela tion to unity. CHAP. II. J ARITHMETIC FRACTIONS. 155 To show how many are taken. 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. 160. It was observed,* that two things are TWO things . necessary to necessary to the clear apprehension 01 an mte- apprehends , number. ger number. 1st. A distinct apprehension of the unit which First. forms the basis of the number ; and, 2dly. A distinct apprehension of the number second. of times which that unit is taken. Three things are necessary to the distinct ap- Three things . necessary to prehension of the value of any fraction, either apprehend a , . , , fraction. 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 second. that unit is divided ; and, 3dly. We must know how many such parts Third. are taken in the expression. The unit from which the fraction is derived, unit of tho fraction of is called the unit of the fraction ; and one of the expres- the equal parts is called, the unit of the expres sion. For example, to apprehend the value of the Section 110. 156 MATHEMATICAL SCIENCE. [BOOK II. What we fraction f of a pound avoirdupois, or jib. ; we must know. must know, First 1st. What is meant by a pound ; second. 2d. That it has been divided into seven equal parts; and, Third. 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, the unit of the expression ; and 3 denotes that three frac tional units are taken. Umt when If the unit of a fraction be not named, it is not named. t , taken to be the abstract unit 1. ADVANTAGES OF FRACTIONAL UNITS. Every equal 161. By considering every equal part of uni- partofone, a unit. ty as a unit of 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. Advantages Under this searching analysis, the mind at once seizes on the unit f the fraction as the principal basis. It then looks at the value of each part. It then inquires how many such parts are taken. Tt havin been shown that e ql integer units tegraiorfrac-can alone be added, it is readily seen that the CHAP. II.] ARITHMETIC ADVANTAGES. 157 same principle is equally applicable to frac- tionai,can tional units ; and then the inquiry is made : added. What is necessary in order to make such units equal ? It is seen at once, that two things are neces- Two thin ? 3 necessary for sary : addition. 1st. That they be parts of the same unit ; and, Fir9t - 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. 162. The great difficulty in the management Difficult y in of fractions, consists in comparing them with ment of frac tions. eack other, instead of constantly comparing them with the unity from which they are derived. By considering them as entire things, having a How fixed relation to the unity which is their basis, obviated, they can be compared as readily as integer num bers ; for, the mind is never at a loss when it apprehends the unit, the parts into which it is divided, and the number of parts which are taken. The only reasons why we apprehend and MATHEMATICAL SCIENCE. [BOOK II. handle integer numbers more readily than frac tions, are, 1st. Because the unity forming the basis is always kept in view ; and, 2d. Because, in integer numbers, we have been taught to trace constantly the connection between the unity and the numbers which come from it ; while in the methods of treating frac tions, these important considerations have been neglected. SECTION III. PROPORTION AND RATIO. 163. PROPORTION expresses the relation which one number bears to another, with respect to its being greater or less. Two numbers may be compared, the one wilii the other, in two ways : it method. 1st. With respect to their difference, called Arithmetical Proportion ; and, 2d method. 2d. With respect to their quotient, called Geometrical Proportion. CHAP. II. J ARITHMETIC PROPORTION. 159 Thus, if we compare the numbers 1 and 8, Example of by their difference, we find that the second ex- ^^ 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 Ratio. ratio. If we compare the same numbers by their Examp j e of Geometrical Proportion. quotient, we find that the second contains the Geometrical first 8 times : hence, 8 is the measure of their geometrical proportion, and is called their geo metrical ratio.* 164. The two numbers which are thus corn- 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 difference, the question is, how much is one y 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 quotient. quotient, the question is, how 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 only in this sense. 1GO MATHEMATICAL SCIENCE. [fiOOK II. Example by of their proportion. Ten, for example, is 9 difference. greater than 1, if 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, hls t " s r e m the that, the units of our common system of numbers 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- convenient ient language to express a particular relation of two numbers, and is perfectly correct, when used in conformity to an accurate definition. in what 165. All authors agree, that the measure of all authors , . agree- the geometrical proportion, between two num bers, is their ratio ; but they are by no means in what disa- unan imous, 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 tliods. standard the y the ""* * A11 a g ree that th e standard, what- divisor, ever it may be, should be made the divisor. * The Encyclopedia Metropolitana, a work distinguished by the excellence of its scientific articles, adopts the latter method. CHAP. II.] ARITHMETIC RATIO. 161 This leads us to inquire, whether the mind what is the best form. 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. 166. All our ideas of numbers begin at origin of numbers. one.* This is the starting-point. We con ceive of a number only by measuring it with How we con one, as a standard. One is primarily in the ^^ mind before we acquire an idea of any other number. Hence, then, the comparison begins Where lhe at one, which is the standard or unit, and all 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 The idea is, how many times does eight contain the stand- P rcsented - ard? We measure by this standard, and the ratio is standard. Ratio. the result of the measurement. In this view of the case, the standard should be the first number ^J^ 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 being the Example. number of times which 6 contains 2. * Section 104. 11 102 MATHEMATICAL SCIENCE. [BOOK II. other reasons 167. The reason for adopting this method for this me thod of com- of comparison will appear still stronger, if we take 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 onej anc j t m s j s 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 168. It may be proper here to observe, that proportion. while the term "geometrical proportion" is used to express the relation of two numbers, com- A geometri- pared by their ratio, the term, " A geometrical CM! propor tion defined, proportion," 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- 169. We will now state some further ad vantages. vantages which result from regarding the ratio as the quotient of the second term divided by the first. o Every <l uestion in ^ Rule of Three is a Three: geometrical proportion, excepting only, that the CHAP. II.] ARITHMETIC - RATIO. 163 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 finding the fourth term, cannot be given, unless Thisrule de - pends on the we define ratio to be the quotient of the second definition of Ratio. term divided by the first. Convenience, there fore, as well as general analogy, indicates this as the proper definition of the term ratio. 170. Again, all authors, so far as I have consulted them, are uniform in their definition tl of the ratio of a geometrical progression : viz. 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 : 16 : 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- ond term by the first : or any other term by the preceding term. But a geometrical progression diners from a geometrical proportion only in 164 MATHEMATICAL SCIENCE. [fiOOK II. The same this : in the former, the ratio of any two terms place m every is the same , while in the latter, the ratio of the first and second is different from that of the sec- au the same. ond and tnird There is, therefore, no essential difference in the two proportions. Why, then, should we say that in the propor tion 2 : 4 : : 6 : 12, the ratio is the quotient of the first term divided Examples. 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 have examined, all the authors authors , , . _ have depart- wno nav e denned the ratio of two numbers to lefimuolf be the q uoti ent 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 m- Most of them have also departed from their stances in , . . . , which the definition, m informing us that " numbers in- deflnition of r crease from right to left in a tenfold ratio," in CHAP. II.] ARITHMETIC PROPORTION. 165 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 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 Thedeflni- all authors, so far as I know, have given their sanction ; although some, it is true, have also used it in a different sense. 171. One important remark on the subject important . Remark. of proportion is yet to be made. It is this : Any two numbers which are compared togeth- Numbers compared er, either by their difference or quotient, must must be of . . . , the same be of the same kind: that is, they must either kind> have the same unit, as a basis, or be susceptible of reduction to the same unit. For example, we can compare 2 pounds with Examples relating to 6 pounds : their difference is 4 pounds, and their Arithmetical ratio is the abstract number 3. We can also rical 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- 166 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 yards of cloth, for they are quantities of different kinds, not being susceptible of reduc tion to a common unit. Abstract Simple or abstract numbers may always be numbers may be compared, compared, since they have a common unit 1 SECTION IY. APPLICATIONS OF THE SCIENCE OF ARITHMETIC. 172. ARITHMETIC is both a science and an Arithmetic: art. It is a science in all that relates to the In what a science, properties, laws, and proportions of numbers. The science is a collection of those connected science de- processes which develop and make known the laws that regulate and govern all the operations performed on numbers. what the 173. Arithmetic is an art, in this : the sci- science per forms, ence lays open the properties and laws of num bers, and furnishes certain principles from which CHAP. II.] ARITHMETIC APPLICATIONS. 1G7 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 art consists. judicious and skilful application of the princi ples of the science ; and the rules contain the directions for such application. 174. In explaining the science of Arithmetic, in explaining great care should be taken that the analysis of wh a t Tilc"^a- every question, and the reasoning by which the principles are proved, be made according to the strictest rules of mathematical logic. Every principle should be laid down and ex- HOW each plained, not only with reference to its subsequent ^oukfbe use and application in arithmetic, but also, with stated> reference to its connection with the entire mathe matical science of which, arithmetic is the ele mentary branch. 175. That analysis of questions, therefore, what , . , . questions are where cost is compared with quantity, or quan- feul ty. 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 yards cost at the same rate ? 1G8 MATHEMATICAL SCIENCE. [BOOK II. Analysis: Analysis. 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, 6 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. , a cnila. Me 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. 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 : analysis : 6 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 as simple as the first, it is certainly mote strictly tiflc. CHAP. II.] ARITHMETIC APPLICATIONS. 169 scientific ; and when once learned, can be ap- its plied through the whole range of mathematical science. 176. There is yet another view of this ques- Reasons in , . i . | , . c favor of the tion which removes, .to a great degree, if not erst 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 Numbe rs existing as between the numbers 6 and 12, also mustbere - garded as ab- regarded as abstract. Now, by leaving out of stract: view, for a moment, the units of the numbers, and finding 12 as an abstract number, and then The analysis ... ., , then correct. assigning to it its proper unit, we have a correct analysis, as well as a correct result. 177. It should be borne in mind, that practi- How the rules of arith- 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 What practical knowledge, it cares nothing about the ^^ e forms or methods of investigation it demands demands. 170 MATHEMATICAL SCIENCE. L BOOK H - the fruits of them all, in the most concentrated Best rule of and practical form. Hence, the best rule of art, ^ which is the one most easily applied, and which reaches the result by the shortest process, is not always constructed after those methods which science employs in the development of its prin ciples. Definition of For example, the definition of multiplication is, mi tion! 1Ca tnat it i tne process of taking one number, called the multiplicand, as many times as there are what it de- units in anotner called the multiplier. This defi- mands. nition, as one of science, requires two things. First. 1st. That the multiplier be an abstract num ber; and, second. 2dly. That the product be of the same kind as the multiplicand. These two principles are certainly correct, Maybe and relating to arithmetic as a science, are uni- versa Uy true - But are they universally true, in the sense in which the 7 would be understood by learners, when applied to arithmetic as a mixed subject, that is, a science and an art ? Such an application would certainly exclude a large class of practical rules, which are used in the appli cations of arithmetic, without reference to par ticular units. Examples of For example> if we haye ^ in ^^ ^ ^ applications, multiplied by feet in height, we must exclude the CHAP. II.] ARITHMETIC APPLICATIONS. 171 question as one to which arithmetic is not appli cable ; or else we must multiply, as indeed we do, without reference to the unit, and then assign a proper unit to the product. If we have a product arising from the three w* 160 the three factors factors of length, breadth, and thickness, the are lines, unit of the first product and the unit of the final product, will not only be different from each other, but both of them will be different from the unit of the given numbers. The unit of the The different given numbers will be a unit of length, the unit of the first product will be a square, and that of the final product, a cube. 178. Again, if we wish to find, by the best other J examples. practical rule, the cost of 467 feet of boards at 30 cents per foot, we should multiply 467 by 30, and declare the cost to be 14010 cents, or $140,10. Now, as a question of science, if you ask, can considered , . , (, , . i as a question we multiply feet by cents f we answer, certainly ofscience< not. If you again ask, is the result obtained right ? we answer, yes. If you ask for the analy- sys, we give you the following : 1 foot of boards : 467 feet : : 30 cents : Answer. Now, the ratio of 1 foot to 467 feet, is the ab- Ratio, stract number 467 ; and 30 cents being multi- 172 MATHEMATICAL SCIENCE. [BOOK II. plied by this number, gives for the product 14010 cents. But as the product of two numbers is Product of two numerically the same, whichever number be used as the multiplier, we know that 467 multiplied by 30, gives the same number of units as 30 multi- The first rule p\i e d by 467 : hence, the first rule for finding the correct amount is correct. scientific in- 179. I have given these illustrations to point out the difference between a process of scientific Practica investigation and a practical rule. The first should always present the ideas of Their difler- the subject in their natural order and connection, what it con- while the other should point out the best way of obtaining a desired result. In the latter, the steps of the process may not conform to the or der necessary for the investigation of principles ; but the correctness of the result must be suscepti ble of rigorous proof. Much needless and un- Causesof P r fitable discussion has arisen on many of the error. processes of arithmetic, from confounding a princi ple of science with a rule of mere application. CHAP. II.] ARITHMETI C ORDER. 173 SECTION Y. METHODS OF TEACHING ARITHMETIC CONSIDERED. ORDER OF THE SUBJECTS. > 180. IT has been well remarked by Cousin, the great French philosopher, that " As is the method of a philosopher, so will be his system ; and the adoption of a method decides the destiny of a philosophy." What is said here of philosophy in general, is eminently true of the philosophy of mathematical science ; and there is no branch of it to which the remark applies, with greater force, than to that of arithmetic. It is here, that the first no tions of mathematical science are acquired. It is here, that the mind wakes up, as it were, to the consciousness of its reasoning powers. Here, it acquires the first knowledge of the abstract separates, for the first time, the pure ideal from the actual, and begins to reflect and reason on pure mental conceptions. It is, therefore, of the highest importance that these first thoughts be impressed on the mind in their natural and proper Cousin. Method decides Philosophy. True in science. Why important in Arithmetic. First thoughts should be rightly impressed. 174 MATHEMATICAL SCIENCE. [BOOK II. Faculties to order, so as to strengthen and cultivate, at the be cultivated. ^^ ^^ the faculties o f apprehension, discrim ination, and comparison, and also improve the yet higher faculty of logical deduction. Firet point: 181. The first point, then, in framing a course of arithmetical instruction, is to deter- methodof mine the method of presenting the subject. Is th^bjed!. there any thing in the nature of the subject it self, or the connection of its parts, that points out the order in which these parts should be Laws of studied ? Do the laws of science demand a whLTdo particular order ; or are the parts so loosely they require? connected) ag to render it a matte r of indiffer- ence where we begin and where we end ? A review of the analysis of the subject will aid us in this inquiry. Basis of the 182. We have seen* that the science of science of . numbers, numbers is based on the unit 1. Indeed, the in what the whole science consists in developing, explain- consists. ing, and illustrating the laws by which, and through which, we operate on this unit. There Three classes are three classes of operations performed on the of operations. unit one. e uie 1st. To increase it according to certain scales, unit. * Section 104. CHAP. II.] ARITHMETIC - INTEGER UNITS. 175 forming the classes of simple and denominate numbers ; 2d. To divide it in any way we please, form- 2d. TO , , . , , , c . divide it. ing the decimal and vulgar fractions ; and, 3d. To compare it with all the numbers which 3d. TO com- come from it ; and then those numbers with each other. This embraces proportions, of which the Rule of Three is the principal branch. There is yet a fourth branch of arithmetic ; Fourth viz. the application of the principles and of the rules drawn from them, in the mechanic arts Practical . ,. . f. applications; and in the ordinary transactions of business. This is called the Art, or practical part, of these the Arithmetic. (See Arithmetical Diagram facing page 117.) Now, if this analysis be correct, it establishes Analysis . , . f, . , . establishes the order in which the subjects of arithmetic the order. should be taught. INTEGER UNITS. 183. We begin first with the unit 1, and in- Unitone crease it according to the scale of tens, forming JJJJJ*- 1 ^ the common system of integer numbers. We the scale of tens. then perform on these numbers the operations of the five ground rules ; viz. numerate them, operations add them, subtract them, multiply and divide * them. 176 MATHEMATICAL SCIENCE. [BOOK II. Next increase We next increase the unit 1 according to the it according to varying varying scales of the denominate numbers, and thus produce the system, called Denominate or Concrete Numbers ; after which we perform upon this class all the operations of the five ground rules. What order 184. It may be well to observe here, that the law of , , exact science tne ^ aw * exact science requires us to treat the denominate numbers first, and the numbers of the common system afterwards; for, the corn- Reason for mon system is but a variety of the class of de- this. nominate numbers ; viz. that variety, in which the scale is the scale of tens, and unvarying. Reason for But as some knowledge of a subject must precede departing . from it. <M generalization, we are obliged to begin the subject of arithmetic with the simplest element. FRACTIONAL UNITS. Divisions of 185. We now pass to the second class of the unit. operations on the unit 1 ; viz. the divisions of General me- it. Here we pursue the most general method, and divide it arbitrarily ; that is, into any num ber of equal parts. We then observe that the Method ao- division of it, according to the scale of tens is cording to , scale of tens. but a particular case of the general law of di vision. We then perform, on the fractional CHAP. II.] ARITHMETIC RATIO. 177 units which thus arise, all the operations of the operations performed. rive ground rules. RATIO, OR RULE OF THREE. 186. Having considered the two subjects of subjects considered. integer and fractional units, we come next to the comparison of numbers with each other. This branch of arithmetic develops all the what this . . . branch do- relative properties of numbers, resulting from velops . their inequality. The method of arrangement, indicated above, what the ar . rangement presents all the operations of arithmetic in con- does . nection with the unit 1, which certainly forms the basis of the arithmetical science. Besides, this arrangement draws a broad line what it does between the science of 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 8 houidbe that the learner shall clearly perceive what is sepan theory and what practice, is of the highest im portance. Teaching things separately, teaching Golden rules , . .for teaching. them well, and pointing out their connections, are the golden rules of all successful instruc tion. 187. I had supposed, that the place of the 12 178 MATHEMATICAL SCIENCE. [BOOK II. Rule of Three, among the branches of arith metic, had been fixed long since. But several Differences in authors of late, have placed most of the practi- cal subjects before this rule giving precedence, for example, to the subjects of Percentage, In- in what they terest, Discount, Insurance, &c. It is not easy consist. . to discover the motive ot this change. It is Ratio part 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 tions, 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 in Percentage, Interest, Discount, Insurance, &c., cannot wen by a separate analysis. But this is a matter ch tinge th6 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 subjects, it is a great aid and a pow- of first ex- c , .,. plaining the crtul auxiliary in explaining and establishing TnL . f a11 the Poetical rules. If the Rule of 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- J^ipTof eral P rinci P le > as early as possible, and then con- induction, nect with it, and with each other, all the subor- CHAP. II.] ARITHMETIC - PRACTICAL PART. 179 dinate principles, with their applications, which flow from it. PRACTICAL PART. 188. We come next to the 4th division; of arithmetic. viz. the applications of arithmetic. Under the classification which we have indi- Whathaa been done. cated, all the principles of the science will have been mastered, when the pupil reaches this stage of his progress. His business will now be with What remains to be the application of principles, and no longer in done. the study and development of the principles themselves. The unity and simplicity of this ^^^ method of classification, may be made more ap parent, by the aid of the arithmetical diagram which faces page 117. May we not then conclude that the subjects Hwthesub- * jects should of arithmetic should be presented in the follow- be presented. ing order : 1st. All the methods of treating integer num- lst - Inte g er numbers. bers, whether formed from the unit 1 according to the scale of tens, or according to varying scales ; 2d. All the methods of treating fractional uni- 2d. Frac tions. ties, whether derived from the unit 1 according to the scale of tens, or according to varying scales ; 180 MATHEMATICAL SCIENCE. [BOOK II. 3d. Rule of 3d. The proportion and ratios of numbers; Three. and, 4th.A PP iica- 4th. The applications of the science of num bers to practical and useful objects. OBJECTIONS TO THIS CLASSIFICATION ANSWERED. 189 - I* nas been urged that Common or Vul- S ar Fractions should be placed "immediately after Division, for two reasons." " First, they arise from division, being in fact unexecuted division." "Second, in Reduction and the Compound Second. Rules, it is often necessary to multiply and divide fractions, to add and subtract them, also to carry for them, unless perchance the examples are con structed for the occasion, and with special refer ence to avoiding these difficulties." These are aii. These, I believe, are all the objections that have been, or can be urged against the classifi cation which I have suggested. I give them in Given in fuii. full, because I wish the subject of arrangement to be fully considered and discussed. It should what be our main object to get at the best possible should be /!/ our object. s y s tem ol classification, and not to waste our efforts in ingenious arguments in the support of TO be con- a favorite one. We will consider these obiec- aidered se parately, tions separately. CHAP. II.] ARITHMETIC OBJECTIONS. 181 It is certainly true, that fractions " arise from Fractions arise from di- division, but it is as certainly not true, that they vision. are " unexecuted divisions ;" and this last idea has involved the subject in much perplexity and difficulty. The most elementary idea of a fraction, arises The element ary idea is from the division of a single thing into two equal obtained by .parts, each of which is called a half. Now, we get no idea of this half unless we consider the division perfected. And indeed, the method of teaching shows this. For, we cannot impress the idea of a half on the mind of a child, until Example; we have actually divided in his presence the apple (or something else regarded as a unit), and exhibited the parts separately to his senses ; and all other fractions must be learned by a like reference to the unit 1. Hence, we can form no And not . . otherwise. notion of a fraction, except on the supposition ot a perfected division. If the term, "unexecuted division," applies to "Unexecuted division"does the numerator of the expression, and not to the not apply to ,. , ,, . i . -, ..II the numera- unit of the fraction, the idea is still more m- tor volved. For, nothing is plainer than that we can form no distinct notion of a result, so long as the process on which it depends cannot be executed. The vague impression that there is That a frao- something hanging about a fraction that cannot tion cannot be quite reached, has involved the subject in a rcacAed , ha s 182 MATHEMATICAL SCIENCE. [BOOK II. occasioned mysterious terror ; and the boy approaches it sulty with the same feeling which a mariner does a rocky and dangerous coast, of which he has neither map nor chart to guide him. But pre sent to the mind of the pupil the distinct idea, that a fraction is one or more equal parts of unity, and that every such part is a perfect whole, axed relation ^ av i n rr a certain relation to the thing from which to unity. it was derived, and all the mist is cleared away, and his mind divides the unit into any number of equal parts, with the same facility as the knife divides the apple. Form the The form of expression for a fraction, and for wTunexecu- an unexecuted division, is indeed the same, but ted division. ^ interpretation of this expression, as used for one or the other, is entirely different. In our A sign may common language, the same word is not al- express dif ferent things, ways the sign of the same idea ; and in science, the same symbol often expresses very different things, Example For example, |^, as an expression in fractions, illustrating heseprinci- means, that something regarded as a wnoie has been divided in 7 equal parts, and that 3 of those parts are taken. As a result of division, it means that the integer number 3 is to be divided into what cannot 7 equal parts. Now, it cannot be assumed, as a be assumed. self-evident fact, that three of the parts of the first division are equal to 1 part of the second ; CHAP. II.] ARITHMETIC - OBJECTIONS. 183 and if this fact be made the basis of a system of fractions, the mind of a child will go through The ^isof every system that system in the dark. The basis of every sys- should be HH elementary tern should be an elementary idea. idea. 190. The second objection, as far as it goes, is valid. In all the tables of denominate num bers, fractions occur five times ; viz. twice in Long Measure, where 5^ yards make 1 rod, and 69J statute miles 1 degree ; once in Cloth Mea sure, where 2-j- inches make 1 nail; once in Square Measure, where 30J square yards make 1 square rod ; and once in Wine Measure, where 31^ gallons make 1 barrel. Now, it were a little better, if these tables had been constructed with integer units. But it should be borne in mind, that the first notions of fractions are given either by oral instruction, or learned from elementary arithmetics. Most of the leading arithmetics are, I believe, preceded by smaller works. These are designed to impart elementary ideas of num- bers, so as not to embarrass the classification of subjects when the scholar is able to enter on a system. Now, the most elementary of these works conducts the pupil, in fractions, far be yond the point necessary to understand and manage all the fractions which appear in the tables of denominate numbers; and hence, there second objec- tion valid * But of no Reasons. Design of wortoi taught in the elementary works; I 184 MATHEMATICAL SCIENCE. [l3OOK II. May then be is no reason, on that account, to depart from a classification otherwise desirable. OBJECTIONS TO THE NEW METHOD. 191. Having examined the objections that have been urged against that system of classifi cation of the subjects of arithmetic, which has Objections to the new me- appeared to me most in accordance with the thod consid- . ered. principles of science, I shall now point out some of the difficulties to be met with in the adoption of the method proposed as a substitute. First objec- 1st. That method separates the simple and de- tiou nominate numbers, which, in their general form ation, differ from each other only in the scale by which we pass from one unit of value to an other, second objeo- 2d. By thus separating these numbers, it be- tion. comes more difficult to point out their connec tion and teach the important fact, that in all their general properties, and in all the opera tions to be performed upon them, they differ from each other in no important particular. Third objec- 3d. By placing the denominate numbers after tion ; T _ Vulgar Fractions, all the principles and rules in limitation of Fractions are limited in their application to a the rules. single class of fractions ; viz. to those fractions which have the same unit CHAP. II.] ARITHMETIC OBJECTIONS. 185 For example, the common rule for addition Examples; of fractions, under this classification, is, in sub stance, the following : " Reduce the fractions to a common denominator; add their numerators, Rule; not and place the sum over the common denomi nator. As the subject of denominate numbers has Have not yet . ii considered not yet been reached, no allusion can be made fractions to fractions having different units. If the learn- er should happen to understand the rule literally, he would conclude that, the sum of all fractions having a common denominator is found by sim ply adding their numerators and placing the The rules therefore ap- sum over the common denominator. But this P i y toone c , . c n -, c class of frac- cannot, of course, be so, since of a and f of a shilling make neither one pound nor one shilling. What appears to me most objectionable in Greatest ob- M i jection. this method, is this : it fails to present the im portant fact, that no two fractions can be blend ed into one, either by addition or subtraction, unless they are parts of the same unit, as we7 as like parts. By this method of classification most of the This method ~ r ofclassifica- difncult questions which arise in fractions are tion avoida avoided, or else the subject must be resumed thedifflcult J questions. after denominate numbers, and that class of questions treated separately. 186 MATHEMATICAL SCIENCE. [fiOOK II. What they The class of questions to which I refer, em braces examples like the following : Add f of a day, T V of an hour, and f of a sec ond together. It is certainly true that a boy will make mar vellous progress in the text-book, if you limit The subject him to those examples in which the fractions posed of, but have a common unit. But, will he ever un- irnt< derstand the science of fractions unless his mind be steadily and always turned to the unit of the fraction, as the basis ? Will he understand the value of one equal part, so as to compare and unite it with another equal part, unless he first apprehends, clearly, the units from which those parts were derived ? Laetobjec- 4th. By placing the Denominate Numbers be- tioD stated. tween Vulgar and Decimal Fractions, the gen eral subject of fractional arithmetic is broken into fragments. This arrangement makes it dif- Difficuityof ficult to realize that these two systems of num- connection of bers differ from each other in no essential par ticular ; that they are both formed from the unit one by the same process, with only a slight mod ification of the scale of division. CHAP. II.] ARITHMETIC LANGUAGE. 187 ARITHMETICAL LANGUAGE. 192. We have seen that the arithmetical al- Arithmetical 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 characters. ten words of the language. If the unit 1 be added to each of the numbers from 1 to 10 in- First ten combina- clusive, we find the first ten combinations in tions. arithmetic.! If 2 be added, in like manner, we have the second ten combinations ; adding second ten, and so on for 3, gives us the third ten combinations; and so others, on, until we have reached one hundred com binations (page 123). Now, as we progressed, each set of combina- Each set giv 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. 193. These one hundred elementary com- AII 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. t Section 116. 188 MATHEMATICAL SCIENCE. [BOOK II. at one hundred, we begin to repeat the numbers, words to be we see that we have but one hundred words to for addition, be remembered for addition; and of these, all only ten above ten are derivative. To this number, uve. must of course be added the few words which express the sums of the hundreds, thousands, &c. subtraction: 194. 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 numbers employed in obtaining them, are words. expressed by twenty words. Muitipiica- 195. In Multiplication (the table being car- tion i ried to twelve), we have one hundred and forty- four elementary combinations,! and fifty-nine Number of separate words (already known) to express the words. results of these combinations. Division: 196. In Division, also, we have one hundred Number of an( * f r ty-fur elementary combinations, f but words. use on ] v t we } ve W ords to express their results. Four hun- 197 Thus, we have four hundred and eigh- tv - ei ht elementary combinations. The results of these combinations are expressed by one hun- ^ re( ^ words ; viz. nineteen in addition, ten in sub- traction fifty-nine in multiplication, and twelve tion, 59 in multi- plication, * Section 120. f Section 122. \ Section 123. CHAP. II.] ARITHMETIC LANGUAGE. 189 in division. Of the nineteen words which are 12 in 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 in 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 different they are the only words employed to translate words in uii. these results from the arithmetical into our com mon language. The language for fractional units is similar Language _ the same for in every particular. By means of a language f rac u OU8 . thus formed we deduce every principle in the science of numbers. 198. Expressing these ideas and their com binations by figures, gives rise to the language Language of arithmetic: of arithmetic. By the aid of 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 figures. 190 MATHEMATICAL SCIENCE. [fiOOK II. But few 199. There is but one further idea to be combinations . . . . . , which presented : it is this, that there are very few sT nufcation com binations made among the figures, which or the figures, change, at all, their signification. Selecting any two of the figures, as 3 and 5, Examples. ^ O1 * exam ple, we see at once that there are but three ways of writing them, that will at all change their signification. First: First, write them by the side of each ) 3 5, other ) 5 3. second: Second, write them, the one over i f, the other J f . Third. 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- Learn the ever, has established that signification, and we language by T ^ 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 its grammar: braces the names of its elementary signs, or Alphabet- Alphabet, the formation and number of its words, and their uses, 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. 191 NECESSITY OF EXACT- DEFINITIONS AND TERMS. 200. The principles of every science are Principles of a collection of mental processes, having estab lished connections with each other. In every branch of mathematics, the Definitions and Definitions and 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 fixed must nifications are once fixed, must always be used always be in the same sense. The necessity of this is most urgent. For, "in the whole range of arithmetical science there is no logical test of truth, but in Reason. a conformity of the reasoning to the definitions and terms, or to such principles as have been established from them." 201. With these principles, as guides, we Definitions ... . . and terms propose to examine some 01 the definitions and exam i ne d. 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. 192 MATHEMATICAL SCIENCE. [BOOK II. It is said, Number de- "Number signifies a unit, or a collection of fined. ., ,, units. HOW " The common method of expressing numbers expressed. .^ ^ ^ Arabic Notation. The Arabic method employs the following ten characters, or figures," &LC. Names of the "The first nine are called significant figures, because each one always has a value, or denotes some number." And a little further on we have, Figures have " The different values which figures have, are values. called simple and local values." The definition of Number is clear and cor- Number rect. It is a general term, comprehending all rightly de- , , . . fined: the phrases which are used, to express, either separately or in connection, one or more things Also figures, of the same 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. , ,, ._ . 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 value ( a mere representative of value. Is it a number or a character to represent number? Is it a It is merely a character: quantity or symbol? It is defined to be a char- CFIAP. II.] ARITHMETIC DEFINITIONS. 193 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- Has no value 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 difficult 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 fig 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. 202. Let us now examine the leading defi- Leading defi nition or principle which forms the basis of the arithmetical language. 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 , T , ~ , . . r . . agree with has been defined as signifying a unit, or a the deflni . collection of units." How, then, can it have a t right hand, or a left ? and how can it increase from right to left in a tenfold ratio?" The explanation given is, that (i each removal of a 13 194 MATHEMATICAL SCIENCE. [BOOK II. Explanation, figure one place towards the left, increases its value ten times" Number, signifying a collection of units, must increase of necessarily increase according to the law by numbers has . . which these units are combined ; and that law ire3 of 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 ideas 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 kind 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. 195 Addition: First. PLACE is merely one of the forms of language Place; by which we designate the unit of a number, its office, expressed by a figure. The definition attributes this property of place both to number and fig ures, while it belongs to neither. 203. Having considered the definitions and terms in the first division of Arithmetic, viz. in notation and numeration, we will now pass to Definitions in 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 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. bers 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 ?" No previous defi nition has been given, in either work, of the 196 MATHEMATICAL SCIENCE. [BOOK II. term SUM. How is the learner to know what he is seeking for, unless that thing be defined ? NO prin- Suppose that John be required to find the sum BbudanL of 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 nition ; 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 Gives a test, process 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. Definitions of 204. I will now quote the definitions of fractions. -p, . Tractions from the same authors, and in the same order of reference. First " 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 - DEFINITIONS. 197 divide or break a whole thing into parts, and that these parts are called fractions, or broken numbers." " Fractions are parts of an integer." second. " When a number or thing is divided into Third. 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, Term fraction , ,, . . defined. means one or more equal parts ol 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 ideas the mind must embrace : 1st. That the thing divided be regarded as a First. standard, or unity ; 2d. That it be divided into equal parts ; second. 3d. That the parts be expressed in terms of Third. the thing divided, regarded as a unit. These ideas are referred to in the latter part of the first definition. Indeed, the definition ^ed: 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 isafrac- ... _. 11- ti n P 81 * f hinted at. Take, for example, the integer num- ^ integer? ber 12, and no one would say that any one part of this number, as 2, 4, or 6, is a fraction. 198 MATHEMATICAL SCIENCE. [BOOK II. Third The third definition would be perfectly accu- rate, by inserting after 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. 205. I have thus given a few examples, illus- Necessity of trating 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- to exactness of thought P ear over-nice, and matters of little moment. e It may be supposed 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 m It is in mathematical science alone that words mathematics. 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. 199 definitions are not such as convey to the mind Must be of the learner, the fundamental ideas of the reason cor- 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 cannot other wise cultivate accurate thinking, 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 W(m isand be strung through the mind on the thread of science, and each have its word or phrase by which it can be transferred to the minds of others. HOW SHOULD THE SUBJECTS BE PRESENTED? 206. Having considered the natural connec- what has been tion of the subjects of arithmetic with each considered, other, as branches of a single science, based on a single unit ; and having also explained the necessity of a perspicuous and accurate lan- * Section 200. 200 MATHEMATICAL SCIENCE. [BOOK II HOW ought guage ; we come now to that important inquiry, the subjects tobepre- How ought those subjects to be presented to the mind of a learner ? Before answering this ques- TWO objects tion, we should reflect, that two important ob- jects should be sought after in the study of arith metic : First. 1st. To train the mind to habits of clear, quick, and accurate thought to teach it to apprehend distinctly to discriminate closely to judge truly and to reason correctly ; and, second. 2d. To give, in abundance, that practical knowledge of the use of figures, in their va rious applications, which shall illustrate the stri- Artofarith- king fact, that the art of arithmetic is the most metic. important art of civilized life being, in fact, the foundation of nearly all the others. now first im- 207. It is certainly true, that most, if not prussions are made. all the elementary notions, whether abstract or practical that is, whether they relate to the science or to the art of arithmetic, must be made on the mind by means of sensible objects. Because of this fact, many have supposed that is reason- the processes of reasoning are all to be con ing to be con- , 11 ducted by ducted by the same sensible objects ; and that objects? ever y abstract principle of science is to be de veloped and established by means of sofas, chairs, apples, and horses. There seems to be CHAP. II.] ARITHMETIC SUBJECTS. 201 an impression that because blocks are useful sensible . , , . . TIT 7 /. objects useful aids in teaching the alphabet, that, therefore m acquiring they can be used advantageously in reading th * 9im P lest J J & elements : Milton and Shakspeare. This error is akin to that of attempting to teach practically, Geog raphy and Surveying in connection with Geom- Error etry, by calling the angles of a rectangle, north, 1^"* south, east, and west, instead of simply designa ting them by the letters A, B, C, and D. This false idea, that every principle of sci- False idea: ence must be learned practically, instead of being rendered practical by its applications, has its effects. been highly detrimental both to science and art. A mechanic, for example, knowing the height Example of his roof and the width of his building, wishes cation of to cut his rafters to the proper length. If he ^^ calls to his aid the established, though abstract principles of science, he finds the length of his rafter, by the well-known relation between the hypothenuse and the two sides of a right-angled triangle. If, however, he will learn nothing ex cept practically, he must raise his rafter to the or learning practically. roof, measure it, and if it be too long cut it on, if too short, splice it. This is the practical way of learning things. The truly practical way, is that in which skill Tme practical. is guided by science. Do the principles above stated find any appli- 202 MATHEMATICAL SCIENCE. [BOOK II. cation in considering the question, How should can arithmetic be taught? Certainly they do. If be applied. . -, arithmetic be both a science and an art, it should be so taught and so learned. Principles 208. The principles of every science are gen eral and abstract truths. They are mere ideas, what primarily acquired through the senses by experi ence, and generalized by processes of reflection wise and reasoning; and when understood, are certain t;> use them. . i i i TIT guides in every case to which they are applicable, If we choose to do without them, w r e may. But is it wise to turn our heads from the guide-boards and explore every road that opens before us ? Now, in the study of arithmetic those princi ples of science, applicable to classes of cases, when should always be taught at the earliest possible and how they should moment. The mind should never be forced through a long series of examples, without ex- The methods planation. One or two examples should always pointed out. precede the statement of an abstract principle, or the laying down of a rule, so as to make the language of the principle or rule intelligible. But to carry the learner forward through a Principles series of them, before the principle on which to be impres sed, they depend has been examined and stated, is forcing the mind to advance mechanically it is lifting up the rafter to measure it, when its CHAP. II.] ARITHMETIC TEXT-BOOKS. 203 exact length could be easily determined by a mle of science. As most of the instruction in arithmetic must Books: be given with the aid of books, we feel unable to do justice to this branch of the subject with- Necessity , for treating out submitting a few observations on the nature of them, of text-books and the objects which they are in tended to answer. TEXT-BOOKS. 209. A text-book should be an aid to the Text-book: teacher in imparting instruction, and to the learner in acquiring knowledge. It should present the subjects of knowledge whatu . ,, should be. in their proper order, with the branches ol each subject classified, and the parts rightly arranged. No text-book, on a subject of general knowledge, selection . ~ . of subjects can contain all that is known ol the subject on nec essary. which it treats ; and ordinarily, it can contain but a very small part. Hence, the subjects to be presented, and the extent to which they are Difficulties , , .,.... of selection. to be treated, are matters of nice discrimination and judgment, about which there must always be a diversity of opinion. 210. The subjects selected should be leading subjects: ones, and those best calculated to unfold, ex- 204 MATHEMATICAL SCIENCE. [flOOK II. plain, and illustrate the principles of the science. HOW They should be so presented as to lead the mind presented. . . . to analyze, discriminate, and classify ; to see each principle separately, each in its combina tion with others, and all, as forming an harmo nious whole. Too much care cannot be be- suggestive stowed in forming the suggestive method of method : arrangement : that is, to place the ideas and principles in such a connection, that each step Reason for. shall prepare the mind of the learner for the next in order. object 211. A text-book should be constructed for of a text- book: the purpose of furnishing the learner with the keys of knowledge. It should point out, explain, Nature; and illustrate by examples, the methods of in vestigating and examining subjects, but should leave the mind of the learner free from the re- straints of minute detail. To fill a book with the analysis of simple questions, which any child can solve in his own way, is to constrain and force the mind at the very point where it is ca pable of self-action. To do that for a pupil, which he can do for himself, is most unwise. detail * 212< A text - book on a subject of science toricai. should not be historical. At first, the minds of children are averse to whatever is abstract, be- CHAP. II.] ARITHMETIC TEXT -BOOKS. 205 cause what is abstract demands thought, and Reasons, thinking is mental labor from which untrained minds turn away. If the thread of science be broken by the presentation of facts, having no connection with the argument, the mind will leave the more rugged path of the reasoning, and employ itself with what requires less effort and labor. The optician, in his delicate experiments, ex- illustration, eludes all light except the beam which he uses : so, the skilful teacher excludes all thoughts excepting those which he is most anxious to impress. As a general rule, subject of course to some exceptions, but one method for each process one method, should be given. The minds of learners should not be confused. If several methods are given, Reasons, it becomes difficult to distinguish the reasonings applicable to each, and it requires much knowl edge of a subject to compare different methods with each other. 213. It seems to be a settled opinion, both HOW the subject is among authors and teachers, that the subject of divided, arithmetic can be best presented by means of three separate works. For the sake of distinc tion, we will designate them the First, Second, and Third Arithmetics. 206 MATHEMATICAL SCIENCE. [BOOK II. We will now explain what we suppose to be the proper construction of each book, and the object for which each should be designed. FIRST ARITHMETIC. First 214. This book should give to the mind Arithmetic : its first direction in mathematical science, and its first impulse in intellectual development. its Hence, it is the most important book of the importance. series. Here, the faculties of apprehension, dis crimination, abstraction, classification and com parison, are brought first into activity. Now, HOW to cultivate and develop these faculties rightly, the subjects must be we must, at first, present every new idea by means of a sensible object, and then immedi ately drop the object and pass to the abstract thought. order We must also present the ideas consecutively; of the ideas. .... that is, in their proper order ; and by the mere method of presentation awaken the comparative and reasoning faculties. Hence, every lesson should contain a given number of ideas. The construction ideas of each lesson, beginning with the first, of the lessons. . should advance in regular gradation, and the lessons themselves should be regular steps in the progress and development of the arithmeti cal science. CHAP. II.] ARITHMETIC TEXT- BOOKS. 207 6 215. The first lesson should merely contain First lesson. representations of sensible objects, placed oppo site names of numbers, to give the impression of the meanings of these names : thus, One * Whatit should con- TWO * * tain. Three * * * &c. &c. And with young pupils, more striking objects should be substituted for the stars. In the second lesson, the words should be re placed by the figures : thus, 1 * 2--------- * * Second 3 #** &c. &c. In the third lesson, I would combine the ideas of the first two, by placing the words and fig ures opposite each other : thus, One - - - - 1 Two .... 2 Three - - - 3 &c. &c. The Roman method of representing numbers should next be taught, making the fourth lesson : viz., Four -.-- 4 Five - --- 5 Third Six - - - - 6 &c. &c. lesson. 208 MATHEMATICAL SCIENCE [BOOK Fourth lesson. One - Two - - - - I. - - - II. Four - - Five - - - IV. - V. Roman method. Three &c. - - III. &c. Six - - &c. - VI. &c. First ten combi- 216. We come now to the first ten cor binations of numbers, which should be given in a separate lesson. In teaching them, we must, of course, have the aid of sensible objects. We teach them thus : One and one are how many ? How taught by things : * One * One * &c. and two are how many ? * * and three are how many ? * * * &c. &c., How in the abstract. through all the combinations : after which, we pass to the abstract combinations, and ask, one and one are how many ? one and two, how many ? one and three, &c. ; after which we express the results in figures. We would then teach in the same manner, in second a separate lesson, the second ten combinations ; tons. then the third, fourth, fifth, sixth, seventh, eighth, ninth, and tenth. In the teaching of these com- wordsused. binations, only the words from one to twenty will have been used. We must then teach the CHAP. II.] ARITHMETIC TEXT-BOOKS. 209 combinations of which the results are expressed Further , , , combina- by the words from twenty to one hundred. tiong Results. How they appear. 217. Having done this, in the way indi cated, the learner sees at a glance, the basis on which the system of common numbers is con structed. He distinguishes readily, the unit one from the unit ten, apprehends clearly how the second is derived from the first, and by com paring them together, comprehends their mutual relation. Having sufficiently impressed on the mind of the learner, the important fact, that numbers are but expressions for one or more things of the same kind, the unit mark may be omitted in the Unit mark combinations which follow. same 218. With the single difference of the omis- sion of the unit mark, the very same method should be used in teaching the one hundred ruje8 * combinations in subtraction, the one hundred and forty-four in multiplication, and the one hundred and forty-four in division. When the elementary combinations of the four ground rules are thus taught, the learner looks Results of , , , , r i the method: back through a series of regular progression, in which every lesson forms an advancing step, and where all the ideas of each lesson have a 14 210 MATHEMATICAL SCIENCE. [BOOK II. mutual and intimate connection with each other. Are they Will not such a system of teaching train the desirable? . r ,. i i mind to the habit 01 regarding each idea sepa- The rately of tracing the connection between each give. new idea and those previously acquired and of comparing thoughts with each other ? and are not these among the great ends to be attained, by instruction ? 219. It has seemed to me of great import- Figures ance to use figures in the very first exercises of should be . , . rT . used early, arithmetic. Unless this be done, the operations must all be conducted by means of sounds, and Reasons, the pupil is thus taught to regard sounds as the proper symbols of the arithmetical language. conse- This habit of mind, once firmly fixed, cannot quences of using words be easily eradicated; and when the figures are learned afterwards, they will not be regarded as the representatives of as many things as their names respectively import, but as the rep resentatives merely of familiar sounds which have been before learned. This would seem to account for the fact, about which, I believe, there is no difference of oral opinion ; that a course of oral arithmetic, ex- arithmetic: 1 . . , tending over the whole subject, without the aid and use of figures, is but a poor preparation for operations on the s^ate. It may, it is true, CHAP. II.] ARITHMETIC TEXT-BOOKS. 211 sharpen and strengthen the mind, and give it what it may do. development: but does it give it that language and those habits of thought, which turn it into what it does not do. the pathways of science? The language of a science affords the tools by which the mind Language . , . , , of arithmetic: pries into its mysteries and digs up its hidden treasures. The language of arithmetic is formed from the ten figures. By the aid of this Ian- its uses, guage we measure the diameter of a spider s web, or the distance to the remotest planet what which circles the heavens ; by its aid, we cal culate the size of a grain of sand and the mag nitude of the sun himself: should we then aban don a language so potent, and attempt to teach its value, arithmetic in one which is unknown in the higher departments of the science ? 220. We next come to the question, how Fractions: the subject of fractions should be presented in an elementary work. The simplest idea of a fraction comes from simplest idea- dividing the unit one into two equal parts. To ascertain if this idea is clearly apprehended, put HOW the question, How many halves are there in one ? The next question, and it is an import- Next TT . question. ant one, is this : How many halves are there in one and one-half? The next, How many halves in two ? How many in two and a half ? In 212 MATHEMATICAL SCIENCE. [BOOK II three ? Three and a half? and so on to twelve. Results. You will thus evolve all the halves from the units of the numbers from one to twelve, in clusive. We stop here, because the multipli cation table goes no further. These combina- First lesson, tions should be embraced in the first lesson on fractions. That lesson, therefore, will teach the rts extent, relation between the unit 1 and the halves, and point out how the latter are obtained from the former. second 221. The second lesson should be the first, reversed. The first question is, how many Grades whole things are there in two halves ? Sec ond, How many whole things in four halves ? How many in eight ? and so on to twenty-four halves, when we reach the extent of the division Extent of table. In this lesson you will have taught the pupil to pass back from the fractions to the unit from which they are derived. Fundamental 222. You have thus taught the two funda mental principles of all the operations in frac tions: viz. First. 1st. To deduce the fractional units from in teger units ; and, second. 2dly. To deduce integer units from fractional units. CHAP . II.] ARITHMETIC TEXT-BOOKS. 213 223. The next lesson should explain the law by which the thirds are derived from the units thirds. from 1 to 12 inclusive ; and the following lesson the manner of changing the thirds into integer units. The next two lessons should exhibit the same Fourths and other operations performed on the fourth, the next fractions. two on the fifth, and so on to include the twelfth. 224. This method of treating the subject of Advantages of the fractions has many advantages : method. 1st. It points out, most distinctly, the relations between the unit 1 and the fractions which are First. derived from it. 2d. It points out clearly the methods of pass- second. ing from the fractional to the integer units. 3d. It teaches the pupil to handle and com- Third. bine the fractional units, as entire things. 4th. It reviews the pupil, thoroughly, through Fourth. the multiplication and division tables. 5th. It awakens and stimulates the faculties Fifth. of apprehension, comparison, and classification. 225. Besides the subjects already named, what T-- 1111 -1 6lSe the Fil St the rirst Arithmetic should also contain the Arithmetic tables of denominate numbers, and collections of simple examples, to be worked on the slate, 214 MATHEMATICAL SCIENCE. [fiOOKII. Examples, under the direction of the teacher. It is not how taught. supposed that the mind of the pupil is suffi ciently matured at this stage of his progress to understand and work by rules. what should be taught in second. Third. Fourth. Fifth. 226. In the First Arithmetic, therefore, , -i i i T i tne PUP" should be taught, 1st. The language of figures; 2d. The four hundred and eighty-eight ele- mentary combinations, and the words by which they are expressed ; 3d. The main principles of Fractions ; 4th. The tables of Denominate Numbers ; and, 5th. To perform, upon the slate, the element ary operations in the four ground rules. second Arithmetic. whatu SECOND ARITHMETIC. 227. This arithmetic occupies a large space . in the school education of the country. Many study it, who study no other. It should, there- fore, be complete in itself. It should also be eminently practical; but it cannot be made so either by giving it the name, or by multiplying the examples. Practical 228. The truly practical cannot be the ante- application of principle, cedent, but must be the consequent of science. CHAP. II.] ARITHMETIC TEXT-BOOKS. 215 Hence, that general arrangement of subjects Arrangement demanded by science, and already explained, must be rigorously followed. But in the treatment of the subjects them- Reasons for . departures. selves, we are obliged, on account of the limited information of the learners, to adopt methods of teaching less general than we could desire. 229. We must here, again, begin with the Basis - unit one, and explain the general formation of the arithmetical language, and must also ad here rigidly to the method of introducing new Method, principles or rules by means of sensible objects. This is most easily and successfully done either How- carried out. by an example or question, so constructed as to show the application of the principle or rule. Such questions or examples being used merely for the purpose of illustration, one or two will Few examples. answer the purpose much better than twenty : for, if a large number be employed, they are Reasons. regarded as examples for practice, and are lost sight of as illustrations. Besides, it confuses the mind to drag it through a long series of examples, before explaining the principles by which they are solved. One example, wrought one example under a rule. under a principle or rule clearly apprehended, conveys to the mind more practical informa tion, than a dozen wrought out as independent 216 MATHEMATICAL SCIENCE. [BOOK II. Principle, exercises. Let the principle precede the prac- Practice. tice, in all cases, as soon as the information acquired will permit. This is the golden rule both of art and morals. subjects 230. The Second Arithmetic should em- embraced. . , . .-. brace all the subjects necessary to a lull view of the science of numbers ; and should contain an abundance of examples to illustrate their practical applications. The reading of numbers, so much (though not too much) dwelt upon, is an invaluable aid in all practical operations. By its aid, in addition, the eye runs up the columns and collects, in a moment, the sum of subtraction: all the numbers. In subtraction, it glances at the figures, and the result is immediately sug gested. In multiplication, also, the sight of the figures brings to mind the result, and it is reached and expressed by one word instead of five. In short division, likewise, there is a cor responding saving of time by reading the results of the operations instead of spelling them. The method of reading should, therefore, be con stantly practised, and none other allowed. Reading : Its value in Addition Multi plication ; Division. CHAP. II.] ARITHMETIC TEXT-BOOKS. 217 THIRD ARITHMETIC. 231. We have now reached the place where Third . , . -T,, Arithmetic: arithmetic may be taught as a science. The pupil, before entering on the subject as treated Preparation here, should be able to perform, at least mechan ically, the operations of the five ground rules. Arithmetic is now to be looked at from an entirely different point of view. The great view of it. principles of generalization are now to be ex plained and applied. Primarily, the general language of figures what ., . is taught must be taught, and the striking tact must then primarily. be explained, that the construction of all integer numbers involves but a single principle, viz. the law of change in passing from one unit to General law: another. The basis of all subsequent operations will thus have been laid. 232. Taking advantage of this general law which controls the formation of numbers, we controls . r formation of bring all the operations 01 reduction under one numbers single principle, viz. this law of change in the unities. Passing to addition, we are equally surprised its value and delighted to find the same principle con trolling all its operations, and that integer num bers of all kinds, whether simple or denominate, may be added under a single rule. 218 MATHEMATICAL SCIENCE. [BOOK II. Advantages This view opens to the mind of the pupil a general law. wide field of thought. It is the first illustra tion of the great advantage which arises from looking into the laws by which numbers are subtraction, constructed. In subtraction, also, the same principle finds a similar application, and a sim ple rule containing but a few words is found applicable to all the classes of integer numbers. In multiplication and division, the same stri king results flow from the same cause; and General thus this simple principle, viz. the law of change law of num- . ,, / 7 7 bers: in passing jrom one unit oj value to another, is the key to all the operations in the four ground rules, whether performed on simple or denomi nate numbers. Thus, all the elementary opera- controis tions of arithmetic are linked to a single prin- cipl e > an d that one a mere principle of arith metical language. Who can calculate the la bor, intellectual and mechanical, which may be saved by a right application of this lumin ous principle ? Design 233. It should be the design of a higher LuhmeticT arithmetic to expand the mind of the learner over the whole science of numbers ; to illus trate the most important applications, and to make manifest the connection between the sci ence and the art. CHAP. II.J ARITHMETIC TEXT-BOOKS. 219 It will not answer these objects if the methods its requisites. of treating the subject are the same as in the elementary works, where science has to com promise with a want of intelligence. An ele mentary is not made a higher arithmetic, by Must have . . -i r . .., a distinctive merely transferring its definitions, its principles, character, and its rules into a larger book, in the same order and connection, and arranging under them an apparently new set of examples, though in fact constructed on precisely the same principles. 234. In the four ground rules, particularly construc- . tion of exain- (where, in the elementary works, simple exam- P ] es inthe pies must necessarily be given, because here they are used both for illustration and practice), the examples should take a wide range, and be so selected and combined as to show thei^ com mon dependence on the same principle. 235. It being the leading design of ft series Design . . i ii . of a series. of arithmetics to explain Hud JIustrate the sci ence and art of numbers, great care should be taken to treat all the subjects, as far as their different natures will permit, according to the same general methods. In passing from one book to another, every subject which has been subjects fully and satisfactorily treated in the one, should f er redwhen be transferred to the other with the fewest pos- fully treated 220 MATHEMATICAL SCIENCE. [BOOK II. HOW com- sible alterations ; so that a pupil shall not have mon subjects maybe to learn under a new dress that which he has already fully acquired. They who have studied the elementary work should, in the higher one, either omit the common subjects or pass them over rapidly in review. The more enlarged and comprehensive views Reasons, which should be given in the higher work will thus be acquired with the least possible labor, and the connection of the series clearly pointed out. This use of those subjects, which have been fully treated in the elementary work, is greatly preferable to the method of attempting to teach Additional every thing anew : for there must necessarily be stated, much that is common ; and that which teaches no new principle, or indicates no new method of application, should be precisely the same in the higher work as in that which precedes it. 236. To vary the examples, in form, without changing in the least the principles on which A contrary they are worked, and to arrange a thousand such method leads to confusion: collections under the same set of rules and sub ject to the same laws of solution, may give a little more mechanical facility in the use of figures, but will add nothing to the stores of arithmetical knowledge. Besides, it deludes the learner with the hope of advancement, and when CHAP. II.] ARITHMETIC CONCLUSION. 221 he reaches the end of his higher arithmetic, he finds, to his amazement, that he has been con It misleads the pupil: ducted by the same guides over the same ground through a winding and devious way, made strange by fantastic drapery : whereas, if what It com plicates the subject. was new had been classed by itself, and what was known clothed in its familiar dress, the sub ject would have been presented in an entirely different and brighter light. CONCLUDING REMARKS. We have thus completed a full analysis of the Conclusion. language of figures, and of the construction of numbers. We have traced from the unit one, all the numbers of arithmetic, whether integer or frac What has been done. tional, whether simple or denominate. We have developed the laws by which they are derived Laws. from this common source, and perceived the connections of each class with all the others. We have examined that concise and beautiful language, by means of which numbers are made Analysis of the lan guage. available in rendering the results of science practically useful ; and we have also considered the best methods of teaching this great subject Methods of teaching indicated. the foundation of all mathematical science and the first among the useful arts. Import ance of the subject. CSIAP. III.] GEOMETRY. 223 CHAPTER III. GEOMETRY DEFINED THINGS OF "WHICH IT TREATS COMPARISON AND PROP ERTIES OF FIGURES DEMONSTRATION PROPORTION SUGGESTIONS FOR TEACHING. GEOMETRY. 237. GEOMETRY treats of space, and com- Geometry, pares portions of space with each other, for the purpose of pointing out their properties and mu tual relations. The science consists in the de- its science, velopment of all the laws relating to space, and is made up of the processes and rules, by means of which portions of space can be best compared with each other. The truths of Geometry are a Its trutns - series of dependent propositions, and may be di- Of three vided into three classes : 1st. Truths implied in the definitions, viz. that 1st. Those implied in things do exist, or may exist, corresponding to thedenni- the words defined. For example : when we say, " A quadrilateral is a rectilinear figure having four sides," we imply the existence of such a figure. 2d. Self-evident, or intuitive truths, embodied ^ A^oms. in the axioms ; and, 3d. Truths inferred from the definitions and 3d. Demon- 224 MATHEMATICAL SCIENCE. [BOOK II. strative axioms, called Demonstrative Truths. We say truths. , ... that a truth or proposition is proved or demon- When de monstrated, strated, when, by a course of reasoning, it is shown to be included under some other truth or proposition, previously known, and from which is said to follow ; hence, Demonstrar A DEMONSTRATION is a Series of logical argU- ments. brought to a conclusion, in which the major premises are definitions, axioms, or prop ositions already established. subjects of 238. Before we can understand the proofs Geometry. or demonstrations of Geometry, we must under stand what that is to which demonstration is applicable : hence, the first thing necessary is to form a clear conception of space, the subject of all geometrical reasoning.* Names of The next step is to give names to particular ms forms or limited portions of space, and to define these names accurately. The definitions of these names are the definitions of Geometry, and the portions of space corresponding to them are Figures, called Figures, or Geometrical Magnitudes ; of Three kinds, which there are three general classes : First. 1st. Lines; second. 2d. Surfaces ; Third. 3d. Solids. * Sections 81 to 85. CHAP. III.] GEOMETRY 2J5 239. Lines embrace only one dimension of Lines, space, viz. length, without breadth or thickness. The extremities, or limits of a line, are called points. There are two general classes of lines straight TWO classes: lines and curved lines. A straight line is one curred.. which lies in the same direction between any two of its points ; and a curved line is one which constantly changes its direction at every point. There is but one kind of straight line, and that is one kind of fully characterized by the definition. From the definition we may infer the following axiom : " A straight line is the shortest distance between two points." There are many kinds of curves, of many of which the circumference of the circle is the sim plest and the most easily described. 240. Surfaces embrace two dimensions of surfaces: space, viz. length and breadth, but not thickness. Surfaces, like lines, are also divided into two pianeand general classes, viz. plane surfaces and curved surfaces. A plane surface is that with which a straight A plane line, any how placed, and having two points common with the surface, will coincide through out its entire extent. Such a surface is per fectly even, and is commonly designated by the Perfectly CY6U* term "A plane." A large class of the figures 15 226 MATHEMATICAL SCIENCE. [BOOK II. piane Fig- of Geometry are but portions of a plane, and all such are called plane figures. 241. A portion of a plane, bounded by three A triangle, straight lines, is called a triangle, and this is the the most sim ple figure, simplest of the plane figures. There are several kinds of triangles, differing from each other, however, only in the relative values of their sides and angles. For example : when the sides are all equal to each other, the triangle is called Kinds of tri- equilateral ; when two of the sides are equal, it is called isosceles ; and scalene, when the three sides are all unequal. If one of the angles is a right angle, the triangle is called a right-angled triangle. 242. The next simplest class of plane figures comprises all those which are bounded by four Quadriiater- straight lines, and are called quadrilaterals. There are several varieties of this class : 1st species. 3 st. The mere quadrilateral, which has no mark, except that of having four sides ; 2d species. 2d. The trapezoid, which has two sides par allel and two not parallel ; 3d species. 3d. The parallelogram, which has its opposite sides parallel and its angles oblique ; 4th species. 4th. The rectangle, which has all its angles right angles and its opposite sides parallel ; and, CHAP. III.] GEOMETRY. 227 5th. The square, which has its four sides equal 5th species. to each other, each to each, and its four angles right angles. 243. Plane figures, bounded by straight lines, other Poiy- having a number of sides greater than four, take names corresponding to the number of sides, viz. Pentagons, Hexagons, Heptagons, &c. 244. A portion of a plane bounded by a circles: curved line, all the points of which are equally distant from a certain point within called the centre, is called a circle, and the bounding line is called the circumference. This is the only the circum- curve usually treated of in Elementary Geometry. 245. A curved surface, like a plane, em- curved sur- faces: braces the two dimensions of length and breadth. It is not even, like the plane, throughout its whole extent, and therefore a straight line may have their proper- ties two points in common, and yet not coincide with it. The surface of the cone, of the sphere, and cylinder, are the curved surfaces treated of in Elementary Geometry. 246. A solid is a portion of space, combi- solids, ning the three dimensions of length, breadth, and thickness. Solids are divided into three classes : Threeciasses. 228 MATHEMATICAL SCIENCE. [BOOK II. 1st class. 1st. Those bounded by planes ; 2d class. 2d. Those bounded by plane and curved sur faces ; and, 3d class. 3d. Those bounded only by curved surfaces, what figures The first class embraces the pyramid and fall in each . . . , . class. prism with their several varieties ; the second class embraces the cylinder and cone ; and the third class the sphere, together with others not generally treated of in Elementary Geometry. Magnitudes 247. We have now named all the geomet rical magnitudes treated of in elementary Ge- what they ometry. They are merely limited portions of space, and do not, necessarily, involve the idea A sphere, of matter. A sphere, for example, fulfils all the conditions imposed by its definitions, without any reference to what may be within the space en- Need not be closed by its surface. That space may be oc- material - j i, i j i_ cupied by lead, iron, or air, or may be a vacuum, without at all changing the nature of the sphere, as a geometrical magnitude. It should be observed that the boundary or Boundaries limit of a geometrical magnitude, is another geo metrical magnitude, having one dimension less. For example : the boundary or limit of a solid, Examples, which has three dimensions, is always a surface which has but two : the limits or boundaries of CHAP. III.] GEOMETRY. 229 all surfaces are lines, straight or curved ; and the extremities or limits of lines are points. 248. We have now named and shown the subjects n timed. nature of the things which are the subjects of Elementary Geometry. The science of Ge- Science f Geometry. ometry is a collection of those connected pro cesses by which we determine the measures, properties, and relations of these magnitudes. COMPARISON OF FIGURES WITH UNITS OF MEASURE. 249. We have seen that the term measure Measure, implies a comparison of the thing measured with some known thing of the same kind, regarded as a standard ; and that such standard is called the unit of measure.* The unit of measure for unitofmeas- lines must, therefore, be a line of a known length : For Lines, a foot, a yard, a rod, a mile, or any other known A Line unit. For surfaces, it is a square constructed Forsurfaces, on the linear unit as a side : that is, a square A square. foot, a square yard, a square rod, a square mile ; that is, a square described on any known unit of length. The unit of measure, for solidity, is a solid, ForSoiids, and therefore has three dimensions. It is a cube A cube. * Section 94. 230 MATHEMATICAL SCIENCE. [BOOK II. constructed on a linear unit as an edge, or on the superficial unit as a base. It is, therefore, a cubic foot, a cubic yard, a cubic rod, &c. Three units Hence, there are three units of measure, each differing in kind from the other two, viz. a known A Line, length for the measurement of lines; a known A square, square for the measurement of surfaces ; and a A cube, known cube for the measurement of solids. The contents: measure or contents of any magnitude, belong- how ascer- ing to either class, is ascertained by finding how many times that magnitude contains its unit of measure. 250. There is yet another class of magni tudes with which Geometry is conversant, called Angles: Angles. They are not, however, elementary magnitudes, but arise from the relative positions Their unit, of those already described. The unit of this class is the right angle ; and with this as a stand ard, all other angles are compared 251. We have dwelt with much detail on the unit of measure, because it furnishes the importance only basis of estimating quantity. The con- of the unit of measure: ception of number and space merely opens to the intellectual vision an unmeasured field of investigation and thought, as the ascent to the summit of a mountain presents to the eye a CHAP. III.] GEOMETRY. 231 wide and unsurveyed horizon. To ascertain the space i i r- i . f IT r i n * te height 01 the point 01 view, the diameter ol the i surrounding circular area and the distance to any point which may be seen, some standard or unity must be known, and its value distinctly apprehended. So, also, number and space, which at first fill the mind with vague and indefinite measured conceptions, are to be finally measured by units by u. of ascertained value. 252. It is found, on careful analysis, that Every num- every number may be referred to the unit one, * as a standard, and when the signification of the the unit one - term ONE is clearly apprehended, that any num ber, whether large or small, whether integer or fractional, may be deduced from the standard by an easy and known process. In space, also, which is indefinite in extent, Space: and exactly similar in all its parts, the faculties of the mind have established ideal boundaries, its ideal These boundaries give rise to the geometrical magnitudes, each of which has its own unit of measure ; and by these simple contrivances, we measure space, even to the stars, as with a yard stick. 253. We have, thus far, not alluded to the difficulty of determining the exact length of that 232 MATHEMATICAL SCIENCE. [fiOOK II. Conception which we regard as a standard. We are pre- o! the unit of measure: sented with a given length, and told that it is called a foot or a yard, and this being usually done at a period of life when the mind is satis fied with mere facts, we adopt the conception At first, a of a distance corresponding to a name, and then ^"sion! 9 " ^ multiplying and dividing that distance we are enabled to apprehend other distances. But this by no means answers the inquiry, What is the standard for measurement ? HOW deter- Under the supposition that the laws of phys- ied * ical nature operate uniformly, the unit of meas ure in England and the United States has been fixed by ascertaining the length of a pendulum which will vibrate seconds, and to this length the Imperial yard, which we have also adopted as a standard, is referred. Hence, the unit of What it is. nieasure is referred to a natural standard, viz. to the distance between the axis of suspension and the centre of oscillation of a pendulum which shall vibrate seconds in vacuo, in London, at the level of the sea. This distance is declared to its length, be 39.1303 imperial inches; that is, 3 imperial feet and 3.1393 inches. Thus, the determina- Difficuities lion of the unit of length demands the applica- bwtt. li n f l ne mos l abstruse science, combined with accurate observation and delicate experiment. Could this distance, or unit, have been exactly CHAP. III.] GEOMETRY. 233 ascertained before the measures of the world were fixed, and in general use, it would have what should . have been anorded a standard at once certain and conve- nient, and all distances would then have been other num- . . . ,, . . hers derived expressed in numbers arising from its multiph- f ro mit. cation or exact division. But as the measures of the world (and consequently their units) were why it is not fixed antecedently to the determination of this distance, it was expressed in measures already known ; and hence, instead of being represented by 1, which had already been appropriated to what now represents it. the foot, it was expressed in terms of the foot, viz. 39.1393 inches, and this is now the standard to which all units of measure are referred. 254. The unit of measure is not only im- unit of meas ure the basis portant as affording a basis for all measurement, of the unit of but is also the element from which we deduce the unit of weight. The weight of 27.7015 cubic inches of distilled water is taken as the standard, weighing exactly one pound avoirdupois, and this quantity of water is determined from the unit of length ; that is, the determination of it reaches what it is. back to the length of a pendulum which will vibrate seconds in the latitude of London. 255. Two geometrical figures are said to be Equivalent equivalent, when they contain the same unit of 234 MATHEMATICAL SCIENCE. [BOOK II. measure an equal number of times. Two figures Equal fig- are said to be equal when they can be so applied to each other as to coincide throughout their Equivalency: whole extent. Hence, equivalency refers to Equality. measure> and equality to coincidence. Indeed, coincidence is the only test of geometrical equal ity. All equal figures are of course equivalent, Their differ- though equivalent figures are by no means equal. Equality is equivalency, with the further mark of coincidence. PROPERTIES OF FIGURES. Property of 256. A property of a figure is a mark cora- figures. mon to all figures of the same class. For exam- pj e : if the class be " Quadrilateral," there are two als. very obvious properties, common to all quadri laterals, besides the one which characterizes the figure, and by which its name is defined, viz. that it has four angles, and that it may be divided into two triangles. If the class be Pwaiieio- " Parallelogram," there are several properties common to all parallelograms, and which are subjects of proof; such as, that the opposite sides and angles are equal ; the diagonals divide each other into equal parts, &c. If the class be Triangle: "Triangle," there are many properties common to all triangles, besides the characteristic that CHAP. III.] GEOMETRY. 235 they have three sides. If the class be a par- Equilateral, ticular kind of triangle, such as the equilateral, isosceles, isosceles, or right-angled triangle, then each class Right-angled. has particular properties, common to every indi vidual of the class, but not common to the other classes. It is important, however, to remark, Every prop erty which that every property which belongs to " triangle," belongs to a , , ... . genus will be regarded as a genus, will appertain to every common to species or class of triangle ; and universally, ev ^ g s . pe ~ every property which belongs to a genus will belong to every species under it ; and every property which belongs to a species will be long to every class or subspecies under it; and a i soto every every property which belongs to one of a sub- species or- class will be common to every indi- individual - vidual of the class. For example : " the square Examples. on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides," is a proposition equally true of every right-angled triangle : and " every straight line perpendicular to a chord, at the circle, middle point, will pass through the centre," is equally true of all circles. MARKS OF WHAT MAY BE PROVED. 257. The characteristic properties of every geometrical figure (that is, those properties with- tlc t P e per 236 MATHEMATICAL SCIENCE. [BOOK II. out which the figures could not exist), are given in the definitions. How are we to arrive at all the other properties of these figures? The propositions implied in the definitions, viz. that Marks: things corresponding to the words defined do or may exist with the properties named ; and the or what may self-evident propositions or axioms, contain the be proved. only marks of what can be proved ; and by a HOW ex- gkjjfui combination of these marks we are able tended. to discover and prove all that is discovered and proved in Geometry. Definitions and axioms, and propositions de- premiss duced ^ rom them, are major premises in each The science: new demonstration; and the science is made up consists. f the processes employed for bringing unfore seen cases under these known truths ; or, in syl logistic language, for proving the minors neces sary to complete the syllogisms. The marks being so few, and the inductions which furnish them so obvious and familiar, there would seem to be very little difficulty in the deductive pro cesses which follow. The connecting together of several of these marks constitutes Deductions, Geometry, or Trains of Reasoning; and hence, Geometry a Deductive TV i * ci science. ls a Deductive Science. CHAP. III.] GEOMETRY. 237 DEMONSTRATION. 258. As a first example, let us take the first proposition in Legendre s Geometry : "If a straight line meet another straight line, Proposition the sum of the two adjacent angles will be equal to two right angles." Let the straight line DC meet the straight line AB Enunciation. at the point C, then will the angle ACD plus the angle DCB be equal to two right AC B angles. To prove this proposition, we need the defini- Thing9 necessary to tion of a right angle, viz. : prove it. When a straight line AB meets another straight line CD, so as to make the ad jacent angles BAG and BAD equal to each other, each of those angles is called a RIGHT ANGLE, and the line AB is said to be PERPENDICULAR to CD. We shall also need the 2d, 3d, and 4th axioms, Axioms, for inferring equality,* viz. : 2. Things which are equal to the same thing second, are equal to each other. f Section 102. 238 MATHEMATICAL SCIENCE. [BOOK II. Third. 3. A whole is equal to the sum of all its parts. Fourth. 4. If equals be added to equals, the sums will be equal. Now before these formulas or tests can be ap- Linetobe plied, it is necessary to sup- E D pose a straight line CE to be Proof: drawn perpendicular to AB at the point C : then by the definition of a right angle, A C B the angle ACE will be equal to the angle ECB. By axiom 3rd, we have, continued: ACD equal to ACE plus ECD : to each of these equals add DCB ; and by the 4th axiom we shall have, ACD plus DCB equal to ACE plus ECD plus DCB ; but by axiom 3rd, ECD plus DCB equals ECB: therefore by axiom 2d, ACD plus DCB equals ACE plus ECB. But by the definition of a right angle, conclusion. ACE plus ECB equals two right angles : there fore, by the 2d axiom, ACD plus DCB equals two right angles, its bases. It will be seen that the conclusiveness of the proof results, First. 1st. From the definition, that ACE and ECB are equal to each other, and each is called a CHAP. III.] GEOMETRY. 239 right-angle : consequently, their sum is equal to two right angles ; and, 2dly. In showing, by means of the axioms, that second. ACD plus DCB equals ACE plus ECB; and then inferring from axiom 2d, that, ACD plus DCB equals two right angles. 259. The difficulty in the geometrical rea- Difficulties in the demon- soning consists mainly in showing that the prop- stations. osition to be proved contains the marks which prove it. To accomplish this, it is frequently necessary to draw many auxiliary lines, forming Auxiliaries new figures and angles, which can be shown to possess marks of these marks, and which thus become connecting links between the known connecting and the unknown truths. Indeed, most of the skill and ingenuity exhibited in the geometrical processes are employed in the use of these auxil iary means. The example above affords an illus tration. We were unable to show that the sum HOW used, of the two angles possessed the mark of being equal to two right angles, until we had drawn a perpendicular, or supposed one drawn, at the point where the given lines intersect. That be ing done, the two right angles ACE and ECB conclusion, were formed, which enabled us to compare the sum of the angle ACD and DCB with two right angles, and thus we proved the proposition. 240 MATHEMATICAL SCIENCE. [BOOK II. Proposition. 260. As a second example, let us take the following proposition : Enunciation. If two straight lines meet each other, the op posite or vertical angles will be equal. Let the straight line AB meet the straight line Diagram. j?T\ r 1 U will the angle ACD be JE equal to the opposite an gle ECB ; and the angle ACE equal to the an gle DCB. Principles To prove this proposition, we need the last isary proposition, and also the 2d and 5th axioms, viz. : " If a straight line meet another straight line, the sum of the two adjacent angles will be equal to two right angles." Axioms. Things which are equal to the same thing are equal to each other." " If equals be taken from equals, the remain ders will be equal." Now, since the straight line AC meets the straight line ED at the point C, we have, proof. ACD plus ACE equal to two right angles. And since the straight line DC meets the straight line AB, we have, ACD plus DCB equal to two right angles : hence, by the second axiom, ACD plus ACE equals ACD plus DCB : ta- CHAP. III.] GEOMETRY. 211 king from each the common angle ACD, we conclusion, know from the fifth axiom that the remain ders will be equal ; that is, the angle ACE equal to the opposite or vertical angle DCB. 261. The two demonstrations given above combine all the processes of proof employed in Demon stra- every demonstration of the same class. When tions generaL any new truth is to be proved, the known tests of truth are gradually extended to auxiliary Useof auxil . quantities having a more intimate connection iaryquar with such new truth than existed between it and the known tests, until finally, the known tests, through a series of links, become applicable to the final truth to be established : the interme diate processes, as it were, bridging over the space between the known tests and the final truth to be proved. 262. There are two classes of demonstra- Direct dera tions, quite different from each other, in some respects, although the same processes of argu mentation are employed in both, and although the conclusions in both are subjected to the same logical tests. They are called Direct, or Ne ^^ Positive Demonstration, and Negative Demon- Reductio ad stration, or the Reductio ad Absurdum. 16 242 MATHEMATICAL SCIENCE. [BOOK II. Difference. 263. The main differences in the two methods are these : The method of direct demon- Direct Dem- stration rests its arguments on known and ad- onstration. mitted truths, and shows by logical processes that the proposition can be brought under some previous definition, axiom, or proposition : while Negative the negative demonstration rests its arguments Demonstra tion, on an hypothesis, combines this with known pro positions, and deduces a conclusion by processes conclusion: strictly logical. Now if the conclusion so de duced agrees with any known truth, we infer ^" h j e h d at that the hypothesis, (which was the only link in the chain not previously known), was true ; but if the conclusion be excluded from the truths previously established ; that is, if it be opposed to any one of them, then it follows that the hy pothesis, being contradictory to such truth, must Determines ^ Q f a } S6i j n the negative demonstration, there- whether the hypothesis is fore, the conclusion is compared with the truths true or false. known antecedently to the proposition in ques tion : if it agrees with any one of them, the hy pothesis is correct ; if it disagrees with any one of them, the hypothesis is false. proof by 264. We will give for an illustration of this Negative emonstra- method, Proposition XVII. of the First Book of Legendre : " When two right-angled triangles have the hypothenuse and a side of the one equal CHAP. III.] GEOMETRY. 243 to the hypothenuse and a side of the other, each Enunciation, to each, the remaining parts will be equal, each to each, and the triangles themselves will be equal." In the two right-angled triangles BAG and EDF (see next figure), let the hypothenuse AC Enunciation be equal to DF, the side BA to the side ED: bythefigure then will the side BC be equal to EF, the angle A to the angle D, and the angle C to the angle F. To prove this proposition, we need the follow ing, which have been before proved ; viz. : Prop. X. (of Legendre). "When two triangles Previous have the three sides of the one equal to the three trutbs neces " sary. sides of the other, each to each, the three an gles will also be equal, each to each, and the triangles themselves will be equal." Prop. V. " When two triangles have two Proposition sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal." Axiom I. " Things which are equal to the Axioms, same thing, are equal to each other." Axiom X. (of Legendre). "All right angles are equal to each other." Prop. XV. " If from a point without a straight Proposition, line, a perpendicular be let fall on the line, and oblique lines be drawn to different points, 1st. " The perpendicular will be shorter than any oblique line ; 244 MATHEMATICAL SCIENCE. [BOOK II. 2d. " Of two oblique lines, drawn at pleasure, that which is farther from the perpendicular will be the longer." Now the two sides BC and Beginning of EF are either equal or un- the demon stration, equal. If they are equal, then by Prop. X. the remain ing parts of the two trian- c G B F E gles are also equal, and the triangles themselves are equal. If the two sides are unequal, one of them must be greater than the other: suppose BC to be the greater. construction On the greater side BC take a part BG, equal e to EF, and draw AG. Then, in the two trian gles BAG and DEF the angle B is equal to the angle E, by axiom X (Legendre), both being right angles. The side AB is equal to the side DE, and by hypothesis the side BG is equal to the side EF : then it follows from Prop. V. that the side AG is equal to the side DF. But the side De uon Stra " DF is e( l ual to the side AC : hence b y axiom *> the side AG is equal to AC. But the line AG cannot be equal to the line AC, having been shown to be less than it by Prop. XV. : hence, conclusion, the conclusion contradicts a known truth, and is therefore false ; consequently, the supposition (on which the conclusion rests), that BC and EF are unequal, is also false ; therefore, they are equal. CHAP. III.] GEOMETRY. 245 265. It is often supposed, though erroneous- Negative ly, that the Negative Demonstration, or the dem- tion \ onstration involving the " reductio ad absurdum," is less conclusive and satisfactory than direct or conclusive, positive demonstration. This impression is sim ply the result of a want of proper analysis. For example : in the demonstration just given, it was Reasons, proved that the two sides BC and EF cannot be unequal; for, such a supposition, in a logi cal argumentation, resulted in a conclusion di- conclusion corresponds redly opposed to a known truth ; and as equality to? or is op _ and inequality are the only general conditions p sedto known truth. of relation between two quantities, it follows that if they do not fulfil the one, they must the other. In both kinds of demonstration, the premises and conclusion agree ; that is, they are Agreement, both true, or both false ; and the reasoning or argument in both is supposed to be strictly logi cal. In the direct demonstration, the premises are known, being antecedent truths ; and hence, the conclusion is true. In the negative demon- Differences in stration, one element is assumed, and the con- k e ir j elusion is then compared with truths previously established. If the conclusion is found to agree with any one of these, we infer that the hy- when the pothesis or assumed element is true ; if it con- tradicts any one of these truths, we infer that 246 MATHEMATICAL SCIENCE. [fiOOK II. tion. when false, the assumed element is false, and hence that its opposite is true. Measured: 266. Having explained the meaning of the term measured, as applied to a geometrical mag nitude, viz. that it implies the comparison of a magnitude with its unit of measure ; and having also explained the signification of the word Prop erty, and the processes of reasoning by which, in all figures, properties not before noticed are inferred from those that are known ; we shall now add a few remarks on the relations of the geometrical figures, and the methods of compar ing them with each other. General Remarks. PROPORTION OF FIGURES. Proportion: 267. Proportion is the relation which one geometrical magnitude bears to another of the same kind, with respect to its being greater or less. The two magnitudes so compared are called its measure, terms, and the measure of the proportion is the quotient which arises from dividing the second Ratio term by the first, and is called their Ratio. Only Quantities of quantities f the same kind can be compared the same together, and it follows from the nature of the kind com pared, relation that the quotient or ratio of any two terms will be an abstract number, whether the terms themselves be abstract or concrete. CHAP. III.] GEOMETRY. 217 268. The term Proportion is defined by most Proportion: , . ,, . i c how defined. authors, "An equality of ratios between lour numbers or quantities, compared together two and two." A proportion certainly arises from such a comparison : thus, if _ = __; then, Example. A U A : B : : C : D is a proportion. But if we have two quantities A and B, which True defim- lion. may change their values, and are, at the same time, so connected together that one of them shall increase or decrease just as many times as the other, their ratio will not be altered by such changes ; and the two quantities are then said JJJf iS. to be in proportion, or proportional. Thus, if A be increased three times and B three times, then, 3 B_A 3A~B ; that is, 3 A and 3 B bear to each other the same proportion as A and B. Science needed a gen- Term need- till eral term to express this relation between two quantities which change their values, without altering their quotient, and the term "propor tional," or " in proportion," is employed for that How used- purpose. 248 MATHEMATICAL SCIENCE. [BOOK II. Reasons for As some apology for the modification of the definition of proportion, which has been so long accepted, it may be proper to state that the term has been used by the best authors in the exact use of the sense here attributed to it. In the definition of term. the second law of motion, we have, " Motion, or change of motion, is proportional to the force impressed ;"* and again, " The inertia of a body is proportioned to its weight."f Similar exam ples may be multiplied to any extent. Indeed, symbol used there is a symbol or character to express the to represent proportion, relation between two quantities, when they un dergo changes of value, without altering their ratio. That character is oc, and is read " pro portional to." Thus, if we have two quantities denoted by A and B, written, Example. A OC B, the expression is read, " A proportional to B." Another kind 269. There is yet another kind of relation of proper- 1-1 i tion. which may exist between two quantities A and B, which it is very important to consider and understand. Suppose the quantities to be so connected with each other, that when the first is increased according to any law of change, the second shall decrease according to the same law ; and the reverse. * Olmsted s Mechanics, p. 28. f Ibid. p. 23. CHAP. III.] GEOMETRY. 249 For example : the area of a rect angle is equal to the product of its base and altitude. Then, in the rectangle ABCD, we have Area = AB x BC. Take a second rectangle EFGH, having a gecond longer base EF, and a less altitude FG, but such Exam P le - that it shall have an equal area with the first : then we shall have Area = EF x FG. Now since the areas are equal, we shall have AB X BC = EF X FG ; Equation. and by resolving the terms of this equation into a proportion, we shall have AB : EF : : FG : BC. Proportion. It is plain that the sides of the rectangle ABCD may be so changed in value as to become the sides of the rectangle EFGH, and that while they are undergoing this change, AB will in crease and BC diminish. The change in the Relations of the Quanti* values of these quantities will therefore take place ties.- according to a fixed law : that is, one will be di minished as many times as the other is increased, 250 MATHEMATICAL SCIENCE. [BOOK II. since their product is constantly equal to the area of the rectangle EFGH. Expressed by Denote the side AB by x and BC by y, and lettera. J J y the area of the rectangle EFGH, which is known, by a; then xy = a; and when the product of two varying quantities is constantly equal to a known quantity, the two Reciprocal quantities are said to be Reciprocally or Inverse- inverse Pro- ty proportional ; thus x and y are said to be in- Ion * versely proportional to each other. If we divide 1 by each member of the above equation, we shall have J__l xy a Redactions and by multiplying both members by x, we shall of the Equations, have 1 X and then by dividing both numbers by x, we have Final form. that is, the ratio of x to - is constantly equal to -; that is, equal to the same quantity, however x or CHAP. III.] GEOMETRY. 251 y may vary ; for, a and consequently - does not change. Hence, Two quantities, which may change their values, Inverse are reciprocally or inversely proportional, when Proportion one is proportional to unity divided by the other, and then their product remains constant. We express this reciprocal or inverse relation thus: Aocl. A is said to be inversely proportional to B : the symbols also express that A is directly propor tional to -Q. If we have o we say, that A is directly proportional to B, and f^ Generally, inversely proportional to G. how rea(L The terms Direct, Inverse or Reciprocal, ap ply to the nature of the proportion, and not to the Ratio, which is always a mere quotient and the measure of proportion. The term Direct ap- Direct and plies to all proportions in which the terms in- te nsno t crease or decrease together ; and the term In- applicable to Ratio. verse or Reciprocal to those in which one term increases as the other decreases. They cannot, therefore, properly be applied to ratio without changing entirely its signification and definition. 252 MATHEMATICAL SCIENCE. [fiOOK II. COMPARISON OF FIGURES. Geometrical 270. In comparing geometrical magnitudes, magnitudes compared, by means of their quotient, it is not the quotient alone which we consider. The comparison im plies a general relation of the magnitudes, which is measured by the Ratio. For example : we Example, say that " Similar triangles are to each other as the squares of their homologous sides." What, do we mean by that ? Just this : Formula of That the area of a triangle Comparison. T _ Is to the area 01 a similar triangle As the area of a square described on a side of the first, To the area of a square described on an ho mologous side of the second. Thus, we see that every term of such a pro- changes of portion is in fact a surface, and that the area how affected f a triangle increases or decreases much faster than its sides ; that is, if we double each side of a triangle, the area will be four times as great: if we multiply each side by three, the area will Results, be nine times as great ; or if we divide each side by two, we diminish the area four times, and so on. Again, circles com- The area of one circle Is to the area of another circle, As a square described on the diameter of the first CHAP. III.] GEOMETRY. 253 To a square described on the diameter of the second. Hence, if we double the diameter of a circle, How their areas change. the area of the circle whose diameter is so in creased will be four times as great : if we mul tiply the diameter by three, the area will be nine times as great ; and similarly if we divide the diameter. 271. In comparing solids together, the same comparison general principles obtain. Similar solids are to each other as the cubes described on their ho- That is, Formula. mologous or corresponding sides. A prism Is to a similar prism, As a cube described on a side of the first, Is to a cube described on an homologous side of the second. Hence, if the sides of a prism be doubled, the How the solidities solid contents will be increased eight-fold. Again, A sphere Is to a sphere, As a cube described on the diameter of the first, Is to a cube described on a diameter of the second. Hence, if the diameter of a sphere be doubled, its solid contents will be increased eight-fold ; if changes. the diameter be multiplied by three, the solid change. Sphere : How its solidity 254 MATHEMATICAL SCIENCE. [fiOOK II. contents will be increased twenty-seven fold : if the diameter be multiplied by four, the solid contents will be increased sixty-four fold ; the solid contents increasing as the cubes of the numbers 1, 2, 3, 4, &c. Ratio : an abstract number. 272. The relation or ratio of two magnitudes to each other, may be, and indeed is, expressed by an abstract number. This number has a fixed value so long as we do not introduce a ing a fixed value. change in the volumes of the solids ; but if we wish to express their ratio under the sup position that their volumes may change ac cording to fixed laws (that is, so that the solids HOW varying s hall continue similar), we then compare them solids are compared, with similar figures described on their homol ogous or corresponding sides; or, what is the same thing, take into account the corresponding changes which take place in the abstract num bers that express their volumes. General outline. Geometry : RECAPITULATION. 273. We have now completed a general outline of the science of Geometry, and what has been said may be recapitulated under the following heads. It has been shown, 1st. That Geometry is conversant about space, CHAP. III.] GEOMETRY. 255 or those limited portions of space which are to what u called Geometrical Magnitudes. 2d. That the geometrical magnitudes embrace three species of figures : 1st. Lines straight and curved ; Lines. 2d. Surfaces plane and curved ; surfaces. 3d. Solids bounded either by plane sur- solids, faces or curved, or both ; and, 4th. Angles, arising from the positions of Angles, lines and planes, and by which they are bounded. 3d. That the science of Geometry is made up science: of those processes by means of which all the up properties of these magnitudes are examined and developed, and that the results arrived at con stitute the truths of Geometry. 4th. That the truths of Geometry may be di- Truths: vided into three classes : three classes. 1st. Those implied in the definitions, viz. First class. that things exist corresponding to certain words defined ; 2d. Intuitive or self-evident truths em- second, bodied in the axioms ; 3d. Truths deduced (that is, inferred) from TtiML the definitions and axioms, called Demonstra tive Truths. 5th. That the examination of the properties of Geometrical the geometrical magnitudes has reference, 256 MATHEMATICAL SCIENCE. [BOOK II. Comparison. 1st. To their comparison with a standard or unit of measure ; Properties. 2d. To the discovery of properties belong ing to an individual figure, and yet common to the entire class to which such figure belongs ; Proportion. 3d. To the comparison, with respect to mag nitude, of figures of the same species with each other ; viz. lines with lines, surfaces with sur faces, and solids with solids. SUGGESTIONS FOR THOSE WHO TEACH GEOMETRY. suggestions. i B e sure t na ^ vour p u pij s have a clear ap- First. prehension of space, and of the notion that Ge ometry is conversant about space only. 2. Be sure that they understand the significa tion of the terms, lines, surfaces, and solids, and that these names indicate certain portions of space corresponding to them. 3. See that they understand the distinction be tween a straight line and a curve; between a plane surface and a curved surface ; between a solid bounded by planes and a solid bounded by curved surfaces. 4. Be careful to have them note the charac- Fourtn. teristics of the different species of plane figures, such as triangles, quadrilaterals, pentagons, hexa gons, &c. ; and then the characteristic of each Third. CHAP. III.] GEOMETRY. 257 class or subspecies, so that the name shall recall, at once, the characteristic properties of each figure. 5. Be careful, also, to have them note the characteristic differences of the solids. Let Firth them often name and distinguish those which are bounded by planes, those bounded by plane and curved surfaces, and those bounded by curved surfaces only. Regarding Solids as a genus, let them give the species and subspecies into which the solid bodies may be divided. 6. Having thus made them familiar with the things which are the subjects of the reasoning, sixth. explain carefully the nature of the definitions ; then of the axioms, the grounds of our belief in them, and the information from which those self-evident truths are inferred. 7. Then explain to them, that the definitions and axioms are the basis of all geometrical rea- g ev enth soning : that every proposition must be deduced from them, and that they afford the tests of all the truths which the reasonings establish. 8. Let every figure, used in a demonstration, be accurately drawn, by the pupil himself, on a Eighth, blackboard. This will establish a connection between the eye and the hand, and give, at the same time, a clear perception of the figure and a distinct apprehension of the relations of its parts. 17 258 MATHEMATICAL SCIENCE. [BOOK II. 9. Let the pupil, in every demonstration, first Ninth, enunciate, in general terms, that is, without the aid of a diagram, or any reference to one, the proposition to be proved ; and then state the principles previously established, which are to be employed in making out the proof. 10. When in the course of a demonstration, Tenth. an y truth is inferred from its connection with one before known, let the truth so referred to be fully and accurately stated, even though the number of the proposition in which it is proved, be also required. This is deemed important. 11. Let the pupil be made to understand that Eleventh, a demonstration is but a series of logical argu ments arising from comparison, and that the result of every comparison, in respect to quan tity, contains the mark either of equality or inequality. 12. Let the distinction between a positive Twelfth, and negative demonstration be fully explained and clearly apprehended. 13. In the comparison of quantities with each Thirteenth, other, great care should be taken to impress the fact that proportion exists only between quan tities of the same kind, and that ratio is the measure of proportion. 14. Do not fail to give much importance to Fourteenth, the kind of quantity under consideration. Let CHAP III,] GEOMETRY. 259 the question be often put, What kind of quantity Fourteenth, are you considering ? Is it a line, a surface, or a solid ? And what kind of a line, surface, or solid ? 15. In all cases of measurement, the unit of measure should receive special attention. If lines are measured, or compared by means of a Fifteenth. common unit, see that the pupil perceives that unit clearly, and apprehends distinctly its rela tions to the lines which it measures. In sur faces, take much pains to mark out on the blackboard the particular square which forms the unit of measure, and write unit, or unit of measure, over it. So in the measurement of solidity, let the unit or measuring cube be ex hibited, and the conception of its size clearly formed in the mind ; and then impress the im portant fact, that, all measurement consists in merely comparing a unit of measure with the quantity measured ; and that the number which expresses the ratio is the numerical expression for that measure. 16. Be careful to explain the difference of the terms Equal and Equivalent, and never permit sixteenth, the pupil to use them as synonymous. An ac curate use of words leads to nice discriminations of thought. CHAP. IV.] ANALY SIS. 261 CHAPTER IV. ANALYSIS ALGEBBA ANALYTICAL GEOMETRY DIFFERENTIAL AND INTEGRAL CALCULUS. ANALYSIS. 274. ANALYSIS is a general term, embra cing that entire portion of mathematical science in which the quantities considered are repre sented by letters of the alphabet, and the opera tions to be performed on them are indicated by signs. Analysis defined. 275. We have seen that all numbers must Numbers must be of be numbers of something;* for, there is no such things; thing as a number without a basis : that is, every number must be based on the abstract unit one, or on some unit denominated. But although numbers must be numbers of something, yet they tut may be of many kuid may be numbers of any thing, for the unit may O f things, be whatever we choose to make it. * Section 112. 262 MATHEMATICAL SCIENCE. [BOOK II. AII quantity 276. All quantity consists of parts, which consists of 1 parts . can be numbered exactly or approximative!}^, and, in this respect, possesses all the properties of number. Propositions, therefore, concerning numbers, have the remarkable peculiarity, that Propositions they are propositions concerning all quantities in regard to number whatever. That half of six is three, is equally apply also to , quantity. true > whatever the word six may represent, whether six abstract units, six men, or six tri angles. Analysis extends the generalization still further. A number represents, or stands for, that particular number of things of the same kind, Algebraic without reference to the nature of the thing ; symbols more gener- but an analytical symbol does more, for it may stand for all numbers, or for all quantities which numbers represent, or even for quantities which cannot be exactly expressed numerically. Anything As soon as we conceive of a thing we may conceived . . ,..,,. may be di- conceive it divided into equal parts, and may Vlded represent either or all of those parts by a or x, or may, if we please, denote the thing itself by a or x, without any reference to its being divided into parts. Each figure 277. In Geometry, each geometrical figure class. a stands for a class ; and when we have demon strated a property of a figure, that property is considered as proved for every figure of the class. CHAP. IV.] ANALYSIS. 263 For example : when we prove that the square Example. described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides, we demonstrate the fact for all right-angled triangles. But in analysis, all numbers, all lines, all surfaces, all in analysis the symbols solids, may be denoted by a single symbol, a or x. stand for Hence, all truths inferred by means of these ^^ symbols are true of all things whatever, and not like those of number and geometry, true only of particular classes of things. It is, therefore, not surprising, that the symbols of analysis do not excite in our minds the ideas of particular Hence, the j truths int er- thmgs. 1 he mere written characters, a, b, c, a, red are geu _ x, y, z, serve as the representatives of things in general, whether abstract or concrete, whether known or unknown, whether finite or infinite. 278. In the uses which we make of these symbols come to be symbols, and the processes of reasoning carried regarded as on by means of them, the mind insensibly comes to regard them as things, and not as mere signs ; and we constantly predicate of them the prop erties of things in general, without pausing to inquire what kind of thing is implied. Thus, Example, we define an equation to be a proposition in Theeq ua- which equality is predicated of one thing as compared with another. For example : 264 MATHEMATICAL SCIENCE. [BOOK II. a -f c = x, whataxioms * s an equation, because x is declared to be necessary to e q U J t O the sum Q f a an( J c J n the Solution of its solution. equations, we employ the axioms, " If equals be added to equals, the sums will be equal ;" and, " If equals be taken from equals, the remainders They express will be equal." Now, these axioms do not ex- qualitiesof things, press qualities 01 language, but properties 01 Hence, in- quantity. Hence, all inferences in mathemat- ferences re- . late to things, ical science, deduced through the instrumentality of symbols, whether Arithmetical, Geometrical, or Analytical, must be regarded as concerning quantity, and not symbols. Quantity As analytical symbols are the representatives tewM- f quantity in general, there is no necessity of enttothe k ee ping the idea of quantity continually alive in the mind ; and the processes of thought may, without danger, be allowed to rest on the sym bols themselves, and therefore, become to that extent, merely mechanical. But, when we look The reason- back and see on what the reasoning is based, and based o/the now tne processes have been conducted, we shall supposition fin( j tnat every ste p was taken on the supposition of quantity. that we were actually dealing with things, and not symbols ; and that, without this understand ing of the language, the whole system is without signification, and fails. CHAP. IV.] ALGEBRA. 265 279. There are three principal branches of Three , . branches : Analysis : 1st. Algebra. Algebra, 2d. Analytical Geometry. Analytical * Geometry, 3d. Differential and Integral Calculus. calculus. ALGEBRA. 280. Algebra is, in fact, a species of uni- Algebra: versal Arithmetic, in which letters and signs are universal employed to abridge and generalize all processes involving numbers. It is divided into two parts, TWO parts: corresponding to the science and art of Arith metic : 1st. That which has for its object the investi- First part: gation of the properties of numbers, embracing all the processes of reasoning by which new properties are inferred from known ones ; and, 2d. The solution of all problems or questions second part, involving the determination of certain numbers which are unknown, from their connection with certain others which are known or given. ANALYTICAL GEOMETRY. 281. Analytical Geometry examines the Analytical , ~ , Geometry. properties, measures, and relations ol the geo metrical magnitudes by means of the analytical its nature. 266 MATHEMATICAL SCIENCE. [BOCK II. symbols. This branch of mathematical science Descartes, originated with the illustrious Descartes, a cele- the original founder of brated French mathematician of the 17th ccn- this science. .. . , . . . c tury. He observed that the positions 01 points, What he observed, the direction of lines, and the forms of surfaces, could be expressed by means of the algebraic AH position symbols ; and consequently, that every change, expressed by symbols, either in the position or extent of a geometrical magnitude, produced a corresponding change in certain symbols, by which such magnitude could be represented. As soon as it was found that. to every variety of position in points, direction in lines, or form of curves or surfaces, there cor responded certain analytical expressions (called their Equations), it followed, that if the processes were known by which these equations could be The equation examined, the relation of their parts determined, develops the properties and the laws according to which those parts Vai 7 relative to one another, ascertained, then the corresponding changes in the geometrical magnitudes, thus represented, could be imme diately inferred. Hence, it follows that every geometrical ques- Power over tion can be solved, if we can resolve the corre- the magni- i i - j ,1 tude extend- sponding algebraic equation ; and the power over edbythe ^ Q geometrical magnitudes was extended iust in equation. proportion as the properties of quantity were brought to light by means of the Calculus. The CHAP. IV.] ANALYSIS. 267 applications of this Calculus were soon made to TO what sub ject applied. the subjects of mechanics, astronomy, and in deed, in a greater or less degree, to all branches of natural philosophy; so that, at the present its present time, all the varieties of physical phenomena, with which the higher branches of the science are conversant, are found to answer to varieties determinate by the algebraic analysis. 282. Two classes of quantities, and conse- Quantities which enter quently two sets of symbols, quite distinct from into the Cai- culus. each other, enter into this Calculus ; the one called Constants, which preserve a fixed or given constants. value throughout the same discussion or investi gation ; and the other called Variables, which variables, undergo certain changes of value, the laws of which are indicated by the algebraic expressions or equations into which they enter. Hence, Analytical Geometry may be defined as that Analytical branch of mathematical science, which exam- defined, ines, discusses, and develops the properties of geometrical magnitudes by noting the changes that take place in the algebraic symbols which represent them, the laws of change being deter mined by an algebraic equation or formula. 208 MATHEMATICAL SCIENCE. [BOOK II. DIFFERENTIAL AND INTEGRAL CALCULUS. Quantities 283 j n fafe b ranc h o f mathematical science, considered. as in Analytical Geometry, two kinds of quan- vanabies, ^ity are considered, viz. Variables and Constants ; Constants. and consequently, two distinct sets of symbols The science, are employed. The science consists of a series of processes which note the changes that take place in the value of the Variables. Those changes of value take place according to fixed laws established by algebraic formulas, and are Marks, indicated by certain marks drawn from the va- riable symbols, called Differential Coefficients. By these marks we are enabled to trace out with the accuracy of exact science the most hidden properties of quantity, as well as the most gen eral and minute laws which regulate its changes of value. Analytical 284. It will be observed, that Analytical and Geometry and the Differential and Integral Cal- caicuius : cu ] us treat o f quantity regarded under the same general aspect, viz. as subject to changes or va- Howthey nations in magnitude according to laws indica- regard quan tity : ted by algebraical formulas; and the quantities, whether variable or constant, are, in both cases, by what represented by the same algebraic symbols, viz. ted the constants by the first, and the variables by CHAP. IV.] ALGEBRA. 269 the final letters of the alphabet. There is, how- Difference; ever, this important difference : in Analytical Geometry all the results are inferred from the in what it m consists. relations which exist between the quantities themselves, while in the Differential and Integral Calculus they are deduced by considering what may be indicated by marks drawn from variable quantities, under certain suppositions, and by marks of such marks. 285. Algebra, Analytical Geometry, the Dif- Analytical Science. ferential and Integral Calculus, extended into the Theory of Variations, make up the subject of analytical science, of which Algebra is the ele mentary branch. As the limits of this work do its parts, not permit us to discuss the subject in full, we shall confine ourselves to Algebra, pointing out, occasionally, a few of the more obvious connec- HOW far tions between it and the two other branches. ALGEBRA. 286. On an analysis of the subject of Alge- Algebra. bra, we think it will appear that the subject itself presents no serious difficulties, and that most of Difficulties. the embarrassment which is experienced by the pupil in gaining a knowledge of its principles, as H ow over- well as in their applications, arises from not at- 270 MATHEMATICAL SCIENCE. [BOOK II. Language, tending sufficiently to the language or signs of the thoughts which are combined in the reason ings. At the hazard, therefore, of being a little diffuse, I shall begin with the very elements of the algebraic language, and explain, with much minuteness, the exact signification of the char- characters acters that stand for the quantities which are the LrtqJnttty! subjects of the analysis ; and also of those signs signs. which indicate the several operations to be per formed on the quantities. Quantities. 287. The quantities which are the subjects HOW divided, of the algebraic analysis may be divided into two classes : those which are known or given, and those which are unknown or sought. The Howrepre- known are uniformly represented by the first letters of the alphabet, a, b, c, d, &c. ; and the unknown by the final letters, x, y, z, v, w, &c. May be in- 288 Quantity is susceptible of being in creased or creasec [ or diminished ;* and there are five oper- diminished. Five opera- ations which can be performed upon a quantity tions: that will give results differing from the quantity itself, viz. : First 1st. To add it to itself or to some other quan tity; * Section 75. CHAP. IV.] ALGEBRA. 271 2d. To subtract some other quantity from it ; second. 3d. To multiply it by a number ; Third. 4th. To divide it ; Fourth. 5th. To extract a root of it. Fifth - The cases in which the multiplier or divisor is 1, are of course excepted ; as also the case Exception, where a root is to be extracted of 1. 289. The five signs which denote these oper- signs, ations are too well known to be repeated here. These, with the signs of equality and inequality, Elements of the the letters of the alphabet and the figures which Algebraic are employed, make up the elements of the alge braic language. The words and phrases of the its words and phrases: algebraic, like those ot every other language, are to be taken in connection with each other, and are not to be interpreted as separate and isolated HOW inter preted, symbols. 5 290. The symbols of quantity are designed symbols of to represent quantity in general, whether abstract or concrete, whether known or unknown; and the signs which indicate the operations to be General. performed on the quantities are to be interpreted in a sense equally general. When the sign plus is written, it indicates that the quantity before Examples, which it is placed is to be added to some other signs plus quantity ; and the sign minus implies the exist- aud minus. 272 MATHEMATICAL SCIENCE. [fiOOK II. ence of a minuend, from which the subtrahend is to be taken. One thing should be observed in signs have regard to the signs which indicate the operations no effect on the nature of that are to be performed on quantities, viz. they a quantity. do not at all affect or change the nature of the quantity before or after which they are written, but merely indicate what is to be done with the Examples: quantity. In Algebra, for example, the minus sign merely indicates that the quantity before which it is written is to be subtracted from in Analytical some other quantity ; and in Analytical Geom- Geometry. etry, that the line before which it falls is esti mated in a contrary direction from that in which it would have been reckoned, had it had the sign o plus ; but in neither case is the nature of the quantity itself different from what it would have been had it had the sign plus, interpreta- The interpretation of the language of Algebra tion of the . , . . . . , language: 1S tne fi rst thing to which the attention of a pupil should be directed ; and he should be drilled on the meaning and import of the symbols, until their significations and uses are as familiar as its necessity, the sounds and combinations of the letters of the alphabet. Elements 291. Beginning with the elements of the explained. , , . language, let any number or quantity be desig nated by the letter a, and let it be required to CHAP. IV.] ALGEBRA. 273 add this letter to itself, and find the result or sum. The addition will be expressed by a + a = the sum. But how is the sum to be expressed ? By simply signification. regarding a as one a, or la, and then observing that one a and one a make two as or 2 a : hence, a + CL =2a; and thus we place a figure before a letter to in dicate how many times it is taken. Such figure is called a Coefficient. Coefficient. 292. The product of several numbers is in- Product: dicated by the sign of multiplication, or by sim ply writing the letters which represent the num bers by the side of each other. Thus, aXbXGXdxf, Or abcdf, how indica ted. indicates the continued product of , b, c, d, and /, and each letter is called a factor of the prod uct : hence, a factor of a product is one of the Factor, multipliers which produce it. Any figure, as 5, written before a product, as 5 abcdf, is the coefficient of the product, and shows that coefficient of the product is taken 5 times. a product. 18 274 MATHEMATICAL SCIENCE. [ BOOK II. ten. Equal fac- 293. If the numbers represented by a, b, c, d, and / were equal to each other, they would what the each be represented by a single letter a, and the becomes, product would then become axaxaxaxa = a 5 ; How that is, we indicate the product of several equal expressed. factors by simply writing the letter once arid placing a figure above and a little at the right of it, to indicate how many times it is taken as Exponent: a factor. The figure so written is called an where writ- exponent. Hence, an exponent is a simple form of expression, to point out how many equal fac tors are employed. Division: 294. The division of one quantity by an- how other is indicated by simply writing the divisor below the dividend, after the manner of a frac tion ; by placing it on the right of the dividend with a horizontal line and two dots between them ; or by placing it on the right with a vertical line between them : thus either form of expression : b Three forms. ~> + &> r \ a indicates the division of b by a. Roots: 295. The extraction of a root is indicated how t ed dica " b y the si s n V - This si s n > when used b y itself indicates the lowest root, viz. the square root. CHAP. IV.] ALGEBRA. 275 If any other root is to be extracted, as the third, fourth, fifth, &c., the figure marking the degree index; of the root is written above and at the left of where writ- xl . ten. the sign ; as, tf~ cube root, tf~ fourth root, &c. The figure so written, is called the Index of the root. We have thus given the very simple and gen- Language eral language by which we indicate every one operations! of the five operations that may be performed on an algebraic quantity, and every process in Al gebra involves one or other of these operations. MINUS S IGN. 296. The algebraic symbols are divided into Algebraic language : two classes entirely distinct from each other, viz. the letters that are used to designate the how divided. quantities which are the subjects of the science, and the signs which are employed to indicate certain operations to be performed on those quantities. We have seen that all the algebraic Algebraic processes: processes are comprised under addition, subtrac- their num- tion, multiplication, division, and the extraction ber. of roots ; and it is plain, that the nature of a DO not quantity is not at all changed by prefixing to it ^uT^rine the sign which indicates either of these opera- ^ uantities - 276 MATHEMATICAL SCIENCE. [fiOOK II. tions. The quantity denoted by the letter a, for example, is the same, in every respect, whatever sign may be prefixed to it ; that is, whether it be added to another quantity, subtracted from it, whether multiplied or divided by any number, or whether we extract the square or cube or any Algebraic other root of it. The algebraic signs, therefore, signs : how regard- must be regarded merely as indicating opera tions to be performed on quantity, and not as affecting the nature of the quantities to which they may be prefixed. We say, indeed, that piu 3 and quantities are plus and minus, but this is an ab- Mmua. breviated language to express that they are to be added or subtracted. Principles of 297. In Algebra, as in Arithmetic and Ge ometry, all the principles of the science are de- Fromwhat duced from the definitions and axioms ; and the deduced. ru | eg or p er f orm i n g the operations are but di rections framed in conformity to such principles. Example. Having, for example, fixed by definition, the power of the minus sign, viz. that any quantity before which it is written, shall be regarded as to be what we subtracted from another quantity, we wish to di scove r the process of performing that subtrac tion, so as to deduce therefrom a general prin ciple, from which we can frame a rule applicable to all similar cases. a c CHAP. IV.] ALGEBRA. 277 SUBTRACTION. 298. Let it be required, for example, to subtraction, subtract from b the difference be- 7 Process. tween a and c. i\ow, having writ ten the letters, with their proper signs, the language of Algebra expresses that it is the difference only between a and c, which is to be taken from b ; and if this difference were Difference, known, we could make the subtraction at once. But the nature and generality of the algebraic symbols, enable us to indicate operations, merely, operations indicated. and we cannot in general make reductions until we come to the final result. In what general way, therefore, can we indicate the true differ ence ? If we indicate the subtraction of a from b, we have b a ; but then b-a b-a Final formula. we have taken away too much from b by the number of units in c, for it was not a, but the difference between a and c that was to be subtracted from b. Having taken away too much, the remainder is too small by c : hence, if c be added, the true remainder will be express ed by b a + c. Now, by analyzing this result, we see that the Analysis of the result. sign of every term of the subtrahend has been changed ; and what has been shown with re- 278 MATHEMATICAL SCIENCE. [BOOK II. Generaiiza- spect to these quantities is equally true of all others standing in the same relation : hence, we have the following general rule for the subtrac tion of algebraic quantities : Change the sign of every term of the subtra- Rule - hend, or conceive it to be changed, and then unite the quantities as in addition. MULTIPLICATION . Multiplies- 299. Let us now consider the case of mul- tion. tiplication, and let it be required to multiply a b by c. The algebraic language expresses signification that the difference between a and b of the a b ac be language, is to be taken as many times as there are units in c. If we knew this difference, we could at once perform the multiplication. But by what gen- Process: eral process is it to be performed without finding that difference ? If we take a, c times, the prod uct will be ac ; but as it was only the difference between a and b, that was to be multiplied by c, its nature, this product ac will be too great by b taken c times ; that is, the true product will be expressed by ac be : hence, we see, that, Principle for If a quantity having a plus sign be multi plied by another quantity having also a plus sign, the sign of the product will be plus ; and CHAP. IV.] A.LGEBRA. 279 if a quantity having a minus sign be multi plied by a quantity having a plus sign, the sign of the product will be minus. 300. Let us now take the most general General case: case, viz. that in which it is required to multi ply a b by c d. Let us again observe that the algebraic lan guage denotes that a b is to be taken as many times as there are units in cd\ ac be and if these two differences a-b c d Its form - ac bc ad-\- bd were known, their product | would at once form the product required. First : let us take a b as many times as there First step. are units in c ; this product, from what has al ready been shown, is equal to ac be. But since the multiplier is not c, but c d, it follows that this product is too large by a b taken d times ; that is, by ad bd: hence, the first prod- second step: uct diminished by this last, will give the true product. But, by the rule for subtraction, this difference is found by changing the signs of the Howtaken - subtrahend, and then uniting all the terms as in addition: hence, the true product is expressed by ac be ad + bd. By analyzing this result, and employing an Analysis of abbreviated language, we have the following gen- 280 MATHEMATICAL SCIENCE. [BOOK II. eral principle to which the signs conform in mul tiplication, viz. : Plus multiplied by plus gives plus in the prod uct ; plus multiplied by minus gives minus ; mi nus multiplied by plus gives minus ; and minus multiplied by minus gives plus in the product. General Principle. Remark. Particular case. Minus sign : Its interpre tation. Form of the product : must be true for quantities of any value. 301. The remark is often made by pupils that the above reasoning appears very satisfac tory so long as the quantities are presented un der the above form ; but why will b multiplied by d give plus bd ? How can the product of two negative quantities standing alone be plus ? In the first place, the minus sign being pre fixed to b and d, shows that in an algebraic sense they do not stand by themselves, but are con nected with other quantities ; and if they are not so connected, the minus sign makes no dif ference ; for, it in no case affects the quantity, but merely points out a connection with other quantities. Besides, the product determined above, being independent of any particular value attributed to the letters a, b, c, and d, must bo of such a form as to be true for all values ; and hence for the case in which a and c are both equal to zero. Making this supposition, the product reduces to the form of + bd. The rules for the signs in division are readily deduced from CHAP. IV.] ALGEBRA. 281 the definition of division, and the principles al- signs m j i ii division. ready laid down. ZERO AND INFINITY. 302. The terms zero and infinity have given zero and rise to much discussion, and been regarded as presenting difficulties not easily removed. It may not be easy to frame a form of language that shall convey to a mind, but little versed in mathe matical science, the precise ideas which these Ideasnot abstruse. terms are designed to express ; but we are un willing to suppose that the ideas themselves are beyond the grasp of an ordinary intellect. The terms are used to designate the two limits of Space and Number. 303. Assuming any two points in space, and joining them by a straight line, the distance be tween the points will be truly indicated by the length of this line, and this length may be ex pressed numerically by the number of times which the line contains a known unit. If now, the points are made to approach each other, the lustration, ,1 c i T MI i- i showing the length of the line will diminish as the points meaning of come nearer and nearer together, until at length, %* when the two points become one, the length of the line will disappear, having attained its limit, 282 MATHEMATICAL SCIENCE. [BOOK II. which is called zero. If, on the contrary, the points recede from each other, the length of the illustration, Ji ne joining them will continually increase ; but showing the meaning of so long as the length of the line can be expressed the term . / infinity. m terms oi a known unit 01 measure, it is not infinite. But, if we suppose the points removed, so that any known unit of measure would occupy no appreciable portion of the line, then the length of the line is said to be Infinite. 304. Assuming one as the unit of number, and admitting the self-evident truth that it may be increased or diminished, we shall have no zero andbL difficulty in understanding the import of the terms zero and infinity, as applied to number. For, if we suppose the unit one to be continually diminished, by division or otherwise, the frac- niustration. tional units thus arising will be less and less, and in proportion as we continue the divisions, they will continue to diminish. Now, the limit or boundary to which these very small fractions Zero: approach, is called Zero, or nothing. So long as the fractional number forms an appreciable part of one, it is not zero, but a finite fraction ; and the term zero is only applicable to that which forms no appreciable part of the standard. Diustration. If, on the other hand, we suppose a number to be continually increased, the relation of this CHAP. IV.] ALGEBRA. 283 number to the unit will be constantly changing. So long as the number can be expressed in terms of the unit one, it is finite, and not infi- infinity; nite; but when the unit one forms no appre ciable part of the number, the term infinite is used to express that state of value, or rather, that limit of value. 305. The terms zero and infinity are there- The terms, fore employed to designate the limits to which employed, decreasing and increasing quantities may be made to approach nearer than any assignable quantity ; but these limits cannot be compared, Are Iimit3 . in respect to magnitude, with any known stand ard, so as to give a finite ratio. 306. It may, perhaps, appear somewhat par- Whylimite? adoxical, that zero and infinity should be defined as " the limits of number and space" when they are in themselves not measurable. But a limit is that " which sets bounds to, or circumscribes ;" Definition of and as all finite space and finite number (and such only are implied by the terms Space and or space and Number), are contained between zero and in finity, we employ these terms to designate the limits of Number and Space. 284 MATHEMATICAL SCIENCE. [fiOOK II. OF THE EQUATION. Deductive 307. We have seen that all deductive rea- reasoning. sonmg involves certain processes of comparison, and that the syllogism is the formula to which those processes may be reduced.* It has also comparison b een stated that if two quantities be compared of quantities. together, there will necessarily result the condi- tion of equality or inequality. The equation is an analytical formula for expressing equality. subject of 30g The su bject of equations is divided equations : how divided, into two parts. The first, consists in finding First part: the equation ; that is, in the process of express ing the relations existing between the quantities considered, by means of the algebraic symbols statement, and formula. This is called the Statement of second part: the proposition. The second is purely deduc tive, and consists, in Algebra, in what is called Solution, the solution of the equation, or finding the value of the unknown quantity ; and in the other branches of analysis, it consists in the discus- Discussion of sion of the equation ; that is, in the drawing out from the equation every thing which it is ca pable of expressing. * Section 98. CHAP. IV.] ALGEBRA. 285 309. Making the statement, or finding the statement: equation, is merely analyzing the problem, and what it is. expressing its elements and their relations in the language of analysis. It is, in truth, col lating the facts, noting their bearing and con nection, and inferring some general law or prin ciple which leads to the formation of an equation. The condition of equality between two quan- Equality of tities is expressed by the sign of equality, which w t ^! is placed between them. The quantity on the HOW ex- left of the sign of equality is called the first mem- lgt member ber, and that on the right, the second member 2d member, of the equation. The first member corresponds to the subject of a proposition ; the sign of equal- subject, ity is copula and part of the predicate, signify- Predicate, ing, is EQUAL TO. Hence, an equation is merely a proposition expressed algebraically, in which Proposition. equality is predicated of one quantity as com pared with another. It is the great formula of analysis. 310. We have seen that every quantity is Abstract. either abstract or concrete :* hence, an equa- concrete, tion, which is a general formula for expressing equality, must be either abstract or concrete. An abstract equation expresses merely the * Section 75. 286 MATHEMATICAL SCIENCE. [BOOK II. relation of equality between two abstract quan tities : thus, Abstract equation. is an abstract equation, if no unit of value be assigned to either member ; for, until that be done the abstract unit one is understood, and the formula merely expresses that the sum of a and b is equal to x, and is true, equally, of all quantities, concrete But if we assign a concrete unit of value, that is, say that a and b shall each denote so many pounds weight, or so many feet or yards of length, x will be of the same denomination, and the equation will become concrete or denominate. rive opera- 311. We have seen that there are five oper- tions maybe performed, ations which may be performed on an algebraic quantity.* We assume, as an axiom, that if the same operation, under either of these pro cesses, be performed on both members of an equation, the equality of the members will not be changed. Hence, we have the five following Axioms. First. AXIOMS. 1. If equal quantities be added to both mem bers of an equation, the equality of the members will not be destroyed. * Section 288. CHAP. IV.] ALGEBRA. 287 2. If equal quantities be subtracted from both second, members of an equation, the equality will not be destroyed. 3. If both members of an equation be multi- Third, plied by the same number, the equality will not be destroyed. 4. If both members of an equation be divided by the same number, the equality will not be destroyed. 5. If the same root of both members of an equation be extracted, the equality of the mem bers will not be destroyed. Every operation performed on an equation will fall under one or other of these axioms, and they afford the means of solving all equations which admit of solution. Fourth. Fifth. Use of axioms. 312. The term Equality, in Geometry, ex- Equality: Its meaning presses that relation between two magnitudes in Geometry. which will cause them to coincide, throughout their whole extent, when applied to each other. The same term, in Algebra, merely implies that its meaning in Algebra. the quantity, of which equality is predicated, and that to which it is affirmed to be equal, contain the same unit of measure an equal num ber of times : hence, the algebraic signification of the term equality corresponds to the signi- corresponds to equiva- fication of the geometrical term equivalency. lency. 288 MATHEMATICAL SCIENCE. [BOOK II. 313. We have thus pointed out some of the marked characteristics of analysis. In Algebra, classes of the elementary branch, the quantities, about quantities in . . . Algebra, which the science is conversant, are divided, as has been already remarked, into known and unknown, and the connections between them, expressed by the equation, afford the means of tracing out further relations, and of finding the values of the unknown quantities in terms of the known. In the other branches of analysis, the quanti- HOW divided ties considered are divided into two general in the other branches of classes, Constant and Variable ; the former pre- Analysis. . / i i serving fixed values throughout the same pro cess of investigation, while the latter undergo changes of value according to fixed laws; and from such changes we deduce, by means of the equation, common principles, and general prop erties applicable to all quantities. correspond- 314. The correspondence between the pro- ence in methods of cesses of reasoning, as exhibited in the subject v of accounted general logic, and those which are employed in for mathematical science, is readily accounted for, when we reflect, that the reasoning process is essentially the same in all cases ; and that any change in the language employed, or in the sub ject to which the reasoning is applied, does not CHAP. IV.] ALGEBRA. 289 at all change the nature of the process, or mate rially vary its form. 315. We shall not pursue the subject of analysis any further; for, it would be foreign to the purposes of the present work to attempt objects of , ,, , the present more than to point out the general features and work: characteristics of the different branches of math ematical science, to present the subjects about which the science is conversant, to explain the peculiarities of the language, the nature of the reasoning processes employed, and of the con necting links of that golden chain which binds extended, together all the parts, forming an harmonious whole. SUGGESTIONS FOR THOSE WHO TEACH ALGEBRA. 1. Be careful to explain that the letters em- Letters are ployed, are the mere symbols of quantity. That 8 y m ^ of, and in themselves, they have no meaning or signification whatever, but are used merely as the signs or representatives of such quantities as they may be employed to denote. 2. Be careful to explain that the signs which signs indi- ,, . cate opera- are used are employed merely for the purpose tions . of indicating the five operations which may be performed on quantity ; and that they indicate 19 SCO MATHEMATICAL SCIENCE. [BOOK II. operations merely, without at all affecting the nature of the quantities before which they are placed. Letters and 3. Explain that the letters and signs are the eieme s of e l ements f tne algebraic language, and that the language, language itself arises from the combination of these elements. Algebraic 4. Explain that the finding of an algebraic formula is but the translation of certain ideas, first expressed in our common language, into the language of Algebra ; and that the interpre ts interpret- tation of an algebraic formula is merely trans ation. lating its various significations into common language. Language. 5. Let the language of Algebra be carefully studied, so that its construction and significa tions may be clearly apprehended. coefficient, 6. Let the difference between a coefficient Exponent and an exponent be carefully noted, and the office of each often explained ; and illustrate ire quently the signification of the language by at tributing numerical values to letters in various algebraic expressions, similar 7. Point out often the characteristics of sim- quantities. ilar and dissimilar quantities, and explain which may be incorporated and which cannot. Minus sign. Q. Explain the power of the minus sign, as shown in the four ground rules, but very par- CHAP. IV.] ALGEBRA. 291 ticularly as it is illustrated in subtraction and multiplication. 9. Point out and illustrate the correspondence between the four ground rules of Arithmetic Arithmetic and Algebra and Algebra; and impress the fact, that their compared, differences, wherever they appear, arise merely from differences in notation and language : the principles which govern the operations being the same in both. 10. Explain with great minuteness and par- Equation, ticularity all the characteristic properties of the its proper ties. equation ; the manner of forming it ; the differ ent kinds of quantity which enter into its com position ; its examination or discussion ; and the different methods of elimination. 11. In the equation of the second degree, be Equation oi careful to dwell on the four forms which em brace all the cases, and illustrate by many ex amples that every equation of the second de gree may be reduced to one or other of them, its forms. Explain very particularly the meaning of the term root ; and then show, why every equation Its roots - of the first degree has one, and every equation of the second degree two. Dwell on the prop erties of these roots in the equation of the sec ond degree. Show why their sum, in all the Their sum. forms, is equal to the coefficient of the second term, taken with a contrary sign ; and why their 292 MATHEMATICAL SCIENCE. [BOOK II. Their prod- product is equal to the absolute term with a uct contrary sign. Explain when and why the roots are imaginary. General \2. In fine, remember that every operation Principles: . ....... , and rule is based on a principle of science, and that an intelligible reason may be given for it. Find that reason, and impress it on the mind should be of your pupil in plain and simple language, and by familiar and appropriate illustrations. You will thus impress right habits of investigation and study, and he will grow in knowledge. The broad field of analytical investigation will be opened to his intellectual vision, and he will have made the first steps in that sublime science They lead to w hich discovers the laws of nature in their most general lawa. secret hiding-places, and follows them, as they reach out, in omnipotent power, to control the motions of matter through the entire regions of occupied space. BOOK III. UTILITY OF MATHEMATICS, CHAPTER I. THE UTILITY OF MATHEMATICS CONSIDERED AS A MEANS OF INTELLECTUAL TRAINING AND CULTURE. 316. THE first efforts in mathematical sci- Firet efforts - ence are made by the child in the process of counting. He counts his fingers, and repeats the words one, two, three, four, five, six, seven, * eight, nine, ten, until he associates with these jects. words the ideas of one or more, and thus ac quires his first notions of number. Hence, the idea of number is first presented to the mind by means of sensible objects ; but when once clear ly apprehended, the perception of the sensible objects fades away, and the mind retains only the abstract idea. Thus, the child, after count- General- tion. ing for a time with the aid of his fingers or his marbles, dispenses with these cumbrous helps, and 294 UTILITY OF MATHEMATICS. [BOOK III. Abstraction, employs only the abstract ideas, which his mind embraces with clearness and uses with facility. Analytical 317. In the first stages of the analytical method: , , . . i i methods, where the quantities considered are uses sensible represented by the letters of the alphabet, sen- first! 3 sible objects again lend their aid to enable the mind to gain exact and distinct ideas of the things considered ; but no sooner are these ideas obtained than the mind loses sight of the things themselves, and operates entirely through the instrumentality of symbols. Geometry. 318. So, also, in Geometry. The right line may first be presented to the mind, as a black First impres- mark on paper, or a chalk mark on a black- sions by sen- . , ., . . sibie objects, board, to impress the geometrical definition, that " A straight line does not change its direction between any two of its points." When this definition is clearly apprehended, the mind needs no further aid from the eye, for the image is forever imprinted. A plane. 319. The idea of a plane surface may be Definition: impressed by exhibiting the surface of a polished mirror; and thus the mind may be aided in HOW mustra- apprehending the definition, that " a plane sur- ted. face is one in which, if any two points be taken, CHAP. I.] QUANTITY SPACE. 295 the straight line which joins them will lie wholly in the surface." But when the definition is understood, the mind requires no sensible object Itetrue conception. to aid its conception. The ideal alone fills the mind, and the image lives there without any connection with sensible objects. 320. Space is indefinite extension, in which space, all bodies are situated. A solid or body is any Solid: portion of space embracing the three dimensions of length, breadth, and thickness. To give to the mind the true conception of a solid, the aid HOW con ceived, of the eye may at first be necessary; but the idea being once impressed, that a solid, in a strictly mathematical sense, means only a por tion of space, and has no reference to the mat- what it ter with which the space may be filled, the mind turns away from the material object, and dwells alone on the ideal. 321. Although quantity, in its general sense, Quantity: is the subject of mathematical inquiry, yet the language of mathematics is so constructed, that Language: the investigations are pursued without the slight- HOW con- structed. est relerence to quantity as a material substance. We have seen that a system of symbols, by which quantities may be represented, has been symbols: adopted, forming a language for the expression 296 UTILITY OF MATHEMATICS. [BOOK III. of ideas entirely disconnected from material ob jects, and yet capable of expressing and repre- Nitureof se nting such objects. This symbolical language, the lan guage: at once copious and exact, not only enables us to express our known thoughts, in every depart- whatitac- ment of mathematical science, but is a potent complishes. ~ , . .... , , means of pushing our inquiries into unexplored regions, and conducting the mind with certainty to new and valuable truths. Advantages 322. The nature of that culture, which the cact mind undergoes by being trained in the use of an exact language, in which the connection be tween the sign and the thing signified is unmis takable, has been well set forth by a living author, greatly distinguished for his scientific attainments.* Of the pure sciences, he says Herschei s " Their objects are so definite, and our no- 3W8< tions of them so distinct, that we can reason about them with an assurance that the words and signs of our reasonings are full and true repre sentatives of the things signified ; and, conse- Exactian- quently, that when we use language or signs in .. argument, we neither by their use introduce extraneous notions, nor exclude any part of the case before us from consideration. For exarn- * Sir John Herschel, Discourse on the study of Natural Philosophy. CHAP. I.] EXACT TERMS. 297 pie : the words space, square, circle, a hundred, Mathematical terms exact. &c., convey to the mind notions so complete in themselves, and so distinct from every thing else, that we are sure when we use them we know and have in view the whole of our own meaning. It is widely different with words ex- Different in . . regard to pressing natural objects and mixed relations. other terms> Take, for instance, Iron. Different persons at tach very different ideas to this word. One who has never heard of magnetism has a widely dif ferent notion of iron from one in the contrary predicament. The vulgar who regard this metal HOW iron is regarded by as incombustible, and the chemist, who sees it the chemist: burn with the utmost fury, and who has other reasons for regarding it as one of the most com bustible bodies in nature ; the poet, who uses The poet it as an emblem of rigidity ; and the smith and engineer, in whose hands it is plastic, and mould ed like wax into every form ; the jailer, who prizes The jailer: it as an obstruction, and the electrician, who The eiectri- sees in it only a channel of open communication by which that most impassable of obstacles, the air, may be traversed by his imprisoned fluid, have all different, and all imperfect notions of the same word. The meaning of such a term Final nius- is like the rainbow everybody sees a different one, and all maintain it to be the same." " It is, in fact, in this double or incomplete L 298 UTILITY OF MATHEMATICS. [fiOOK III. incomplete sense of words, that we must look for the origin meaning the c , ~ , . ... source of a ver y large portion of the errors into which error - we fall. Now, the study of the abstract sciences, Mathematics such as Arithmetic, Geometry, Algebra, &c., such errors, while they afford scope for the exercise of rea soning about objects that are, or, at least, may be conceived to be, external to us; yet, being free from these sources of error and mistake, Requires a accustom us to the strict use of language as strict use of . . . language, an instrument of reason, and by familiarizing us in our progress towards truth, to walk uprightly and straightforward, on firm ground, give us that proper and dignified carriage of mind which Results, could never be acquired by having always to pick our steps among obstructions and loose fragments, or to steady them in the reeling tem pests of conflicting meanings." TWO ways of 323. Mr. Locke lays down two ways of in- acquiring knowledge, creasing our knowledge : 1st. "Clear and distinct ideas with settled names ; and, 2d. " The finding of those which show their agreement or disagreement ;" that is, the search ing out of new ideas which result from the com bination of those that are known. First. In regard to the first of these ways, Mr. Locke says : " The first is to get and settle in our minds CHAP. I.] INCREASING KNOWLEDGE. 299 determined ideas of those things, whereof we ideas of have general or specific names ; at least, of so be distinct< many of them as we would consider and im prove our knowledge in, or reason about." * * * " For, it being evident, that our knowledge can not exceed our ideas, as far as they are either im perfect, confused, or obscure, we cannot expect to have certain, perfect, or clear knowledge." 324. Now, the ideas which make up our why it is . C ir-i so in mathe- knowledge oi mathematical science, mini ex- ma tic3. actly these requirements. They are all im pressed on the mind by a fixed, definite, and certain language, and the mind embraces them as so many images or pictures, clear and dis tinct in their outlines, with names which sug gest at once their characteristics and properties. 325. In the second method of increasing second. our knowledge, pointed out by Mr. Locke, math ematical science offers the most ample and the why mathe- m, . 11 i j matics offer surest means. The reasonings are all based on the suregt self-evident truths, and are conducted by means meaus - of the most striking relations between the known and the unknown. The things reasoned about, and the methods of reasoning, are so clearly apprehended, that the mind never hesitates 01 doubts. It comprehends, or it does not compre- 300 UTILITY OF MATHEMATICS. [BOOK III. hend, and the . line which separates the known characteris- f rom the unknown, is always well defined. These tics of the reasoning, characteristics give to this system of reasoning itsadvan- a superiority over every other, arising, not from any difference in the logic, but from a difference in the things to which the logic is applied. Ob servation may deceive, experiment may fail, and experience prove treacherous, but demonstration tion certain. never. Mathematics " If it be true, then, that mathematics include includes a ~ r . certain sys- a perfect system 01 reasoning, whose premises em are self-evident, and whose conclusions are irre sistible, can there be any branch of science or knowledge better adapted to the improvement of the understanding? It is in this capacity, An adjunct as a strong and natural adjunct and instrument and instru- ,, ... ment of rea- f reason, that this science becomes the fit sub-. lon; ject of education with all conditions of society, whatever may be their ultimate pursuits. Most sciences, as, indeed, most branches of knowledge, address themselves to some particular taste, or subsequent avocation ; but this, while it is be fore all, as a useful attainment, especially adapts itself to the cultivation and improvement of the and necessa- thinking faculty, and is alike necessary to all ^ who would be governed by reason, or live for usefulness."* * Mansfield s Discourse on the Mathematics. CHAP. I.] REASONS. 301 326. The following, among other consider- ations, may serve to point out and illustrate the value of value of mathematical studies, as a means of mathematics - mental improvement and development. 1. We readily conceive and clearly appre- First. They give hend the things of which the science treats ; clear they being things simple in themselves and read- ily presented to the mind by plain and familiar language. For example : the idea of number, of one or more, is among the first ideas implanted Example. in the mind ; and the child who counts his fin gers or his marbles, understands the art of num bering them as perfectly as he can know any thing. So, likewise, when he learns the definition They estab- of a straight line, of a triangle, of a square, of relations be- a circle, or of a parallelogram, he conceives the idea of each perfectly, and the name and the thing8 - image are inseparably connected. These ideas, so distinct and satisfactory, are expressed in the simplest and fewest terms, and may, if necessary, be illustrated by the aid of sensible objects. 2. The words employed in the definitions second. Words are are always used in the same sense each ex- always used ,, , . i , i , in the same pressing at all times the same idea ; so that when a definition is apprehended, the concep tion of the thing, whose name is defined, is per fect in the mind. There is, therefore, no doubt or ambiguity 302 UTILITY OF MATHEMATICS. [BOOK III. Hence, it is either in the language, or in regard to what is affirmed or denied of the things spoken of; but all is certainty, both in the language employed and in the ideas which it expresses. Third. 3. The science of mathematics employs no no definition definition which may not be clearly compre- or axiom not i j j i j identand nen ded lays down no axioms not universally true, and to which the mind, by the very laws of its nature, readily assents ; and because, also, in the process of the reasoning, no principle or truth is taken for granted, but every link in Theconnec- the chain of the argument is immediately con- tion evident. , . , , n . . nected with a definition or axiom, or with some principle previously established. Fourth. 4. The order established in presenting the The order . strengthens subject to the mind, aids the memory at the ev clear. same ti me tnat it strengthens and improves the reasoning powers. For example: first, there HOW ideas are the definitions of the names of the things ire presen,- ^-^ are ^ su bj ects o f the reasoning; then the axioms, or self-evident truths, which, to gether with the definitions, form the basis of the science. From these the simplest propositions HOW the de- are deduced, and then follow others of greater iUC iow Sftl " difficulty; the whole connected together by rig orous logic each part receiving strength and light from all the others. Whence, it follows, that any proposition may be traced to first prin- CHAP. I.] SYNTHESIS ANALYSIS. 303 ciples ; its dependence upon and connection Propositions with those principles made obvious ; and its truth the iTsources. established by certain and infallible argument. 5. The demonstrative argument of mathe- Fifth - Argument matic^ produces the most certain knowledge the most cert aiii* of which the mind is susceptible. It estab lishes truth so clearly, that none can doubt or deny. For, if the premises are certain that is, Reasons. such that all minds admit their truth without hesitation or doubt, and if the method of draw ing the conclusions be lawful that is, in accord ance with the infallible rules of logic, the infer ences must also be true. Truths thus established may be relied on for their verity ; and the knowl- such edge thus gained may well be denominated SCIENCE. 327. There are, as we have seen, in mathe matics, two systems of investigation quite differ ent from each other : the Synthetical and the g yn thesis, Analytical ; the synthetical beginning with the Anal y sis - definitions and axioms, and terminating in the highest truth reached by Geometry. "This science presents the very method by Synthetical which the human mind, in its progress from childhood to age, develops its faculties. What first meets the observation of a child ? Upon Firgt notionfl what are his earliest investigations employed ? 304 UTILITY OF MATHEMATICS. [BOOK III. is first Next to color, which exists only to the sight, observed. figure, extension, dimension, are the first objects which he meets, and the first which he examines. He ascertains and acknowledges their existence ; then he perceives plurality, and begins to enu- Progress of merate ; finally he begins to draw conclusions from the parts to the whole, and makes a law from the individual to the species. Thus, he has obtained figure, extension, dimension, enu meration, and generalization. This is the teach ing of nature ; and hence, when this process Process de- becomes embodied in a perfect system, as it is veloped in . ^ the system of m Geometry, that system becomes the easiest letry and most natural means of strengthening the mind in its early progress through the fields of knowledge." First neces- " Long after the child has thus begun to gen- eranze an d deduce laws, he notices objects and events, whose exterior relations afford no con clusion upon the subject of his contemplation. Machinery is in motion effects are produced, its method. He is surprised ; examines and inquires. He reasons backward from effect to cause. This is Analysis, the metaphysics of mathematics ; and what the through all its varieties in Arithmetic in Alge- 6 cienceis: bra and in the Differential and Integral Calcu lus, it furnishes a grand armory of weapons for acute philosophical investigation. But analysis CHAP, i.] BACON S OPINION. 305 advances one step further by its peculiar nota- what it does-. tion; it exercises, in the highest degree, the fac ulty of abstraction, which, whether morally or intellectually considered, is always connected with the loftiest efforts of the mind. Thus this science comes in to assist the faculties in their progress to the ultimate stages of reasoning; and the more these analytical processes are cul- what it finally ac tivated, the more the mind looks in upon itself, compiishes. estimates justly and directs rightly those vast powers which are to buoy it up in an eternity of future being."* 328. To the quotations, which have already been so ample, we will add but two more. " In the mathematics, I can report no defi- Bacon s , -, opinion of cience, except it be that men do not sum- math ematics. ciently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For, if the wit be too dull, they sharpen it ; if too wandering, they fix it ; if too inherent in the sense, they abstract it."f Again : " Mathematics serve to inure and corroborate HOW the study of the mind to a constant diligence in study, to * Mansfield s Discourses on Mathematics, f Lord Bacon. 20 336 UTILITY OF MATHEMATICS. [BOOK III. mathematics undergo the trouble of an attentive meditation, ^rnind. and cheerfully contend with such difficulties as lie in the way. They wholly deliver us from credulous simplicity, most strongly fortify us against the vanity of skepticism, effectually re- its influences, strain us from a rash presumption, most easily incline us to due assent, perfectly subjugate us to the government and weight of reason, and inspire us with resolution to wrestle against the injurious tyranny of false prejudices. HOW they are "If the fancy be unstable and fluctuating, it is, as it were, poised by this ballast, and steadied by this anchor ; if the wit be blunt, it is sharp ened by this whetstone ; if it be luxuriant, it is pruned by this knife ; if it be headstrong, it is restrained by this bridle ; and if it be dull, it is roused by this spur."* 329. Mathematics, in all its branches, is, in fact, a science of ideas alone, unmixed with mat- Mathematics ter or material things; and hence, is properly a pure sci- terme( j a p u re Science. It is, indeed, a fairy ence. land of the pure ideal, through which the mind is conducted by conventional symbols, as thought is conveyed along wires constructed by the hand of man. * Dr. Barrow. CHAP. I."] CONCLUSION 307 330. In conclusion, therefore, we may claim what may for the study of Mathematics, that it impresses cilhnedfor the mind with clear and distinct ideas ; culti- m vates habits of close and accurate discrimina tion ; gives, in an eminent degree, the power of abstraction ; sharpens and strengthens all the faculties, and develops, to their highest range, the reasoning powers. The tendency of this its tendency, study is to raise the mind from the servile habit of imitation to the dignity of self-reliance and self-action. It arms it with the inherent ener gies of its own elastic nature, and urges it out The reasons. on the great ocean of thought, to make new discoveries, and enlarge the boundaries of men tal effort. ICS UTILITY OF MATHEMATICS. [BOOK III. CHAPTER II. THE UTILITY OF MATHEMATICS REGARDED AS A MEANS OF ACQUIRING KNOWLEDGE BACONIAN PHILOSOPHY. Mathematics: 331. In the preceding chapter, we consid ered the effects of mathematical studies on the mind, merely as a means of discipline and train- How consid- ing. We regarded the study in a single point ered hereto fore: oi view, viz. as the drill-master of the intel lectual faculties the power best adapted to bring them all into order to impart strength, and to give to them organization. In the HOW now present chapter we shall consider the study un- red der a more enlarged aspect as furnishing to man the keys of hidden and precious knowl edge, and as opening to his mind the whole volume of nature. Material 332. The material universe, which is spread Universe. out before us, is the first object of our rational CHAP. II.] MATERIAL UNIVERSE. 309 regards. Material things are the first with which we have to do. The child plays with his toys Elements of in the nursery, paddles in the limpid water, twirls his top, and strikes with the hammer. At a maturer age a higher class of ideas are embraced. The earth is surveyed, teeming with its products, and filled with life. Man looks around him with wondering and delighted eyes, obtained by mi -.I- -i i observation. Ihe earth he stands upon appears to be made of firm soil and liquid waters. The land is broken into an irregular surface by abrupt hills and frowning mountains. The rivers pursue their courses through the valleys, without any course of iKiturc z apparent cause, and finally seem to lose them selves in their own expansion. He notes the return of day and night, at regular intervals, turns his eyes to the starry heavens, and in quires how far those sentinels of the night may be from the world they look down upon. He is yet to learn that all is governed by general Governed laws imparted by the fiat of Him who created ^fwsT all things ; that matter, in all its forms, is sub ject to those laws ; and that man possesses the Man P os- capacity to investigate, develop, and understand them. It is of the essence of law that it in- vestigate and understand eludes all possible contingencies, and insures tnem - implicit obedience; and such are the laws of nature. 310 UTILITY OF MATHEMATICS. [BOOK III. 333. To the man of chance, nothing is more mysterious than the developments of science. Uniformity: He does not see how so great a uniformity can Variety : jonsist with the infinite variety which pervades every department of nature. While no two individuals of a species are exactly alike, the resemblance and conformity are so close, that the naturalist, from the examination of a sin gle bone, finds no difficulty in determining the species, size, and structure of the animal. So, They appear also, in the vegetable and mineral kingdoms : in all things. all the structures of growth or formation, al though infinitely varied, are yet conformable to like general laws. science ne- This wonderful mechanism, displayed in the cessary to ,, the devei- structure of animals, was but imperfectly under- stood, until touched by the magic wand of sci ence. Then, a general law was found to per vade the whole. Every bone is of that length What science and diameter best adapted to its use ; every muscle is inserted at the right point, and works about the right centre ; the feathers of every bird are shaped in the right form, and the curves in which they cleave the air are best adapted what may to velocity. It is demonstrable, that in every be demon strated. case, and in all the variety of forms in which forces are applied, either to increase power or gain velocity, the very best means have been CHAP. II.] PHILOSOPHY OF BACON. 311 adopted to produce the desired result. And why why it is so. should it not be so, since they are employed by the all- wise Architect ? 334. It is in the investigations of the laws Applications of nature that mathematics finds its widest Mathematics, range and its most striking applications. Experience, aided by observation and enlight ened by experiment, is the recognised fountain Bacon s of all knowledge of nature. On this foundation Bacon rested his Philosophy. He saw that the Deductive process of Aristotle, in which the conclusions do not reach beyond the premises, Aristotle s: was not progressive. It might, indeed, improve the reasoning powers, cultivate habits of nice discrimination, and give great proficiency in verbal dialectics ; but the basis was too narrow for that expansive philosophy, which was to its defects, unfold and harmonize all the laws of nature. Hence, he suggested a careful examination of what Bacon nature in every department, and laid the foun- ! dations of a new philosophy. Nature was to be interrogated by experiment, observation .was to note the results, and gather the facts into the storehouse of knowledge. Facts, so obtained, The means to were subjected to analysis and collation, and general laws inferred from such classification by 312 UTILITY OF MATHEMATICS. [iJOOK III. Bacon s a reasoning process called Induction. Hence, inductive. the system of Bacon is said to be Inductive. 335. This new philosophy gave a startling impulse to the human mind. Its subject was Nature material and immaterial ; its object, the discovery and analysis of those general laws what it did. which pervade, regulate, and impart uniformity to all things ; its processes, experience, experi ment, and observation for the ascertainment of its nature, facts analysis and comparison for their classifi cation ; and reasoning, for the establishment of what aided general laws. . But the work would have been incomplete without the aid of deductive science. General laws deduced from many separate cases, what it by Induction, needed additional proof; for, they needed might have been inferred from resemblances too slight, or coincidences too few. Mathematical science affords such proofs. Thetruthsof 336. Regarding general laws, established by lon: Induction, as fundamental truths, expressing these by means of the analytical formulas, and then operating on these formulas by the known pro- How verified cesses of mathematical science, we are enabled, by Analysis. . not only to verify the truths of induction, but often to establish new truths, which were hidden from experiment and observation. As the in- CHAP. II. 1 EXPERIMENTAL SCIENCE. 313 ductive process may involve error, while the deductive cannot, there are weighty scientific reasons, for giving to every science as much of the character of a Deductive Science as pos sible. Every science, therefore, should be con- Asfarns structed with the fewest and simplest possible inductions. These should be made the basis of made Deduc- deductive processes, by which every truth, how- tive - ever complex, should be proved, even if we chose to verify the same by induction, based on specific experiments. 337. Every branch of Natural Philosophy Natural . . ,, . losophy was was originally experimental; each generaliza- expenmen- tion rested on a special induction, and was de rived from its own distinct set of observations and experiments. From being sciences of pure experiment, as the phrase is, or, to speak more correctly, sciences in which the reasonings con- is now ,, deductive. sist of no more than one step, and that a step of induction ; all these sciences have become, to some extent, and some of them in nearly their whole extent, sciences of pure reasoning : thus, multitudes of truths, already known by induc tion, from as many different sets of experiments, have come to be exhibited as deductions, or co- Matiu-mati- rollaries from inductive propositions of a simpler cal or and more universal character. Thus, mechan- l 314 UTILITY OF MATHEMATICS. [fiOOK III. Deductive ics, hydrostatics, optics, and acoustics, have successively been rendered mathematical ; and astronomy was brought by Newton within the laws of general mechanics. Their advan- The substitution of this circuitous mode of proceeding for a process apparently much easier and more natural, is held, and justly too, to be the greatest triumph in the investigation of nature. They rest on But, it is necessary to remark, that although, by Inductions. this progressive transformation, all sciences tend to become more and more deductive, they are not, therefore, the less inductive ; for, every step in the deduction rests upon an antecedent in- sciencesde- duction. The opposition is, perhaps, not so ductive or ex perimental, much between the terms Deductive and Induc tive as between Deductive and Experimental. ^- sc ^ ence i s experimental, in propor- tai science: t j on as every new case, which presents any pe culiar features, stands in need of a new set of observations and experiments, and a fresh in duction. It is deductive, in proportion as it can whende- ^ raw conclusions, respecting cases of a new kiiH^ by processes which bring these cases un der old inductions, or show them to possess known marks of certain attributes. 339. We can now, therefore, perceive, what CHAP. II.] DEDUCTIVE SCIENCES. 315 is the generic distinction between sciences that Difference can be made deductive and those which must, as yet, remain experimental. The difference consists in our having been able, or not yet able, to draw from first inductions as from a general law, a series of connected and depend ent truths. When this can be done, the de ductive process can be applied, and the sci ence becomes deductive. For example : when Deductive. Newton, by observing and comparing the mo tions of several of the heavenly bodies, discov ered that all the motions, whether regular or Example. apparently anomalous, of all the observed bodies of the Solar System, conformed to the law of moving around a common centre, urged by a centripetal force, varying directly as the mass, and inversely as the square of the distance from the centre, he inferred the existence of such a whatNew- lawfor all the bodies of the system, and then de- ton inferred: monstrated, by the aid of mathematics, that no other law could produce the motions. This is what he the greatest example which has yet occurred of P roved - the transformation, at one stroke, of a science which was in a great degree purely experimen tal, into a deductive science. 340. How far the study of mathematics pre- study of pares the mind for such contemplations and mathematic3: 316 UTILITY OF MATHEMATICS. [BOOK III. prepares the such knowledge, is well set forth by an old wri ter, himself a distinguished mathematician. He says : Dr. Barrow s it The, steps are guided by no lamp more clear- opinion. ly through the dark mazes of nature, by no thread more surely through the infinite turnings of the labyrinth of philosophy ; nor lastly, is the bottom of truth sounded more happily by any other line. HOW I w il} not mention with how plentiful a stock mathematics furnish the of knowledge the mind is furnished from these ; with what wholesome food it is nourished, and what sincere pleasure it enjoys. But if I speak further, I shall neither be the only person nor the first, who affirms it, that while the mind is Abstract abstracted, and elevated from sensible matter, and elevate ... it : distinctly views pure torms, conceives the beau ty of ideas, and investigates the harmony of pro portions, the manners themselves are sensibly corrected and improved, the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more confirmed by divine contemplations : all of which I might, de- philosophers. fend by the authority and confirm by the suf frages of the greatest philosophers."* 341. Sir John Herschel, in his Introduction * Dr. Barrow. CHAP. II.] ASTRONOMY WITHOUT MATHEMATICS. 317 to his admirable Treatise on Astronomy, very opinions, justly remarks, that, " Admission to its sanctuary [the science of Mathemat ical science. Astronomy], and to the privileges and feelings indispensa- ~ -i -i i -i We to a of a votary, is only to be gained by one means knowledge of sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, with out which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an inde pendent opinion on any subject of discussion within their range. "It is not without an effort that those who informa- tion cannot possess this knowledge can communicate on be given such subjects with those who do not, and adapt their language and their illustrations to the ne- cessities of such an intercourse. Propositions which to the one are almost identical, are the orems of import and difficulty to the other ; nor is their evidence presented in the same way to the mind of each. In treating such proposi- Except i i , -, , by very cum- tions, under such circumstances, the appeal has brousmetn- to be made, not to the pure and abstract reason, but to the sense of analogy to practice and experience : principles and modes of action have to be established, not by direct argument from acknowledged axioms, but by continually refer ring to the sources from which the axioms them- Reasons: 18 UTILITY OF MATHEMATICS. [BOOK III. Must begin selves have been drawn, viz. examples; that is to piesteie- Sa 7> ^7 bringing forward and dwelling on simple " ts: and familiar instances in which the same prin ciples and the same or similar modes of action take place ; thus erecting, as it were, in each particular case, a separate induction, and con structing at each step a little body of science to illustration meet its exigencies. The difference is that of of the differ ence be- pioneering a road through an untraversed coun- stmctionby ^T anc ^ advancing at ease along a broad and scientific and b ea t en highway i that is to say, if we are deter- unscientific J J methods, mined to make ourselves distinctly understood, and will appeal to reason at all/ Again : Mathematics " A certain moderate degree of acquaintance ^k a ^stract science is highly desirable to every one who would make any considerable progress in physics. As the universe exists in time and place ; and as motion, velocity, quantity, num ber, and order, are main elements of our knowl edge of external things and their changes, an acquaintance with these, abstractedly consid- is so ered (that is to say, independent of any consid eration of particular things moved, measured, counted, or arranged), must evidently be a use ful preparation for the more complex study of nature/ * * Sir John Herschel on the study of Natural Philosophy. II CHAP. II.] ASTRONOMY. 310 342. If we consider the department of chem- Necessary in chemistry. istry, which analyzes matter, examines the ele ments of which it is composed, develops the laws which unite these elements, and also the agencies which will separate and reunite them, we shall find that no intelligent and philosophical analysis can be made without the aid of mathematics. 343. The mechanism of the physical uni- Laws long unknown. verse, and the laws which govern and regulate its motions, were long unknown. As late as the 17th century, Galileo was imprisoned for pro- Galileo, mulgating the theory that the earth revolves on its axis; and to escape the fury of persecution, His theory. renounced the deductions of science. Now, ev ery student of a college, and every ambitious boy NOW known of the academy, may, by the aid of his Algebra and Geometry, demonstrate the existence and operation of those general laws which enable By what means de- him to trace with certainty the path and mo- monstra ted. tions of every body which circles the heavens. 344. What knowledge is more precious, or Va iue more elevating to the mind, than that which assures us that the solar system, of which the sun is the centre, and our earth one of the smaller bodies, is governed by the general law of gravitation ; that is, that each body is re tained in its orbit by attracting, and being at- L_ 320 UTILITY OF MATHEMATICS. [BOOK III. what tracted by, all the others ? This power of attrac- it teaches. , , . . . . tion, by which matter operates on matter, is the great governing principle of the material world. The motion of each body in the heavens de- The things pends on the forces of attraction of all the not easy. , i r- others ; hence, to estimate such lorces varying as they do with the quantity of matter in each body, and inversely as the squares of their dis tances apart is no easy problem ; yet analy- Anaiysis: sis has solved it, and with such certainty, that the exact spot in the heavens may be marked at which each body will appear at the expiration What it has of any definite period of time. Indeed, a tele- done: scope may be so arranged, that at the end of HOW a that time either one of the heavenly bodies be verified would present itself to the field of view ; and by ^ t en " if the instrument could remain fixed, though the time were a thousand years, the precise mo ment would discover the planet to the eye of the observer, and thus attest the certainty of science. 345. But analysis has done yet more. Tt has not only measured the attractive power of Analysis eac h of the heavenly bodies ; determined their determines balancing distances from a common point and from each forces. other; ascertained their specific gravities and traced their orbits through the heavens ; but has also discovered the existence of balancing CHAP. II.] STABILITY OF THE UNIVERSE. 3J1 and conservative forces, evincing the highest Evidence of evidence of contrivance and design. 346. A superficial view of the architecture of the heavens might inspire a doubt of the sta bility of the entire system. The mutual action of the bodies on each other produces what is called an irregularity in their motions. The earth, for example, in her annual course around the sun, is affected by the attraction of the moon and of all the planets which compose the solar system ; and these attracting forces appear to give an irregularity to her motions. The moon in her revolutions around the earth is also influenced by the attraction of the sun, the earth, and of all the other planets, and yields to each a motion exactly proportionate to the force exert ed ; and the same is equally true of all the bodies which belong to the system. It was reserved for analysis to demonstrate that every supposed irregularity of motion is but the consequence of a general law ; that every change is constancy, and every diversity uniformity. Thus, mathe matical science assures us that our system has not been abandoned to blind chance, but that a superintending Providence is ever exerted through those general laws, which are so minute as to govern the motions of the feather as it is 21 Architecture of the heav ens shows permanency. Example of the earth : Of the moon. Of the other planets. Mathematics proves the permanency of the sys tem. 822 UTILITY OF MATHEMATICS. [BOOK III. so Generality of wafted along on the passing breeze, and yet s omnipotent as to preserve the stability of worlds 347. But analysis goes yet another step. That class of wandering bodies, known to us by comets: the name of comets, although apparently escaped from their own spheres, and straying heedlessly what through illimitable space, have yet been pursued mathematics J proves in re- by the telescope of the observer until sufficient gard to them. data have been obtained to apply the process of analysis. This done, a few lines written upon paper indicate the precise times of their reap- Rosuits stri- pearance. These results, when first obtained, were so striking, and apparently so far beyond the reach of science itself, as almost to need Verification, the verification of experience. At the appointed times, however, the comets reappear, and sci ence is thus verified by observation. Nature 348. The great temple of nature is only to cannot be in- ... vestigated be opened by the keys of mathematical science. ma y P erna P s reach the vestibule, and gaze with wonder on its gorgeous exterior and its exact proportions, but we cannot open the por- niustration. tal and explore the apartments unless we use the appointed means. Those means are the exact sciences, which can only be acquired by discipline and severe mental labor. CHAP. II.] RESULTS OF SCIENCE. 323 The precious metals are not scattered pro- science: fusely over the surface of the earth ; they are, for wise purposes, buried in its bosom, and can be disinterred only by toil and labor. So with science : it comes not by inspiration ; it is not borne to us on the wings of the wind ; it can only to be acquired by neither be extorted by power, nor purchased by 8ludy : wealth ; but is the sure reward of diligent and assiduous labor. Is it worth that labor ? What Tt fc worth study. is it not worth ? It has perforated the earth, and she has yielded up her treasures ; it has it has done guided in safety the bark of commerce over dis- f or the wanta . of man : tant oceans, and brought to civilized man the treasures and choicest products of the remotest climes. It has scaled the heavens, and searched out the hidden laws which regulate and govern the material universe ; it has travelled from planet to planet, measuring their magnitudes, sur veying their surfaces, determining their days and nights, and the lengths of their seasons. It has also pushed its inquiries into regions of space, what i t f. i it nu9 done where it was supposed that the mind of the to make us Omnipotent never yet had energized, and there ^" located unknown worlds calculating their di- verse - ameters, and their times of revolution. 349. Mathematical science is a magnetic Ho w telegraph, which conducts the mind from orb mathematlcs 324 UTILITY OF MATHEMATICS. [BOOK III. aid the to orb through the entire regions of measured mind in its inquiriea: space. It enables us to weigh, in the balance of universal gravitation, the most distant planet of the heavens, to measure its diameter, to de termine its times of revolution about a common centre and about its own axis, and to claim it as a part of our own system. HOW they In these far Teachings of the mind, the im- enlarge it: agination has full scope for its highest exercise. It is not led astray by the false ideal and fed by illusive visions, which sometimes tempt reason from her throne, but is ever guided by the de- May be ductions of science ; and its ideal and the real are united by the fixed laws of eternal truth. Mind 350. There is that within us which delights certainty. m certainty. The mists of doubt obscure the mental, as the mists of the morning do the phys ical vision. We love to look at nature through a medium perfectly transparent, and to see every object in its exact proportions. The science of why mathematics is that medium through which the mathematics afford it. mind may view, and thence understand all the parts of the physical universe. It makes man ifest all its laws, discovers its wonderful harmo nies, and displays the wisdom and omnipotence of the Creator. CHAP. HI.] "PRACTICAL." 325 CHAPTER III. THE UTILITY OF MATHEMATICS CONSIDERED AS FURNISHING THOSE RULES OF ART WHICH MAKE KNOWLEDGE PRACTICALLY EFFECTIVE. 351. THERE is perhaps no word in the Eng- practical: lish language less understood than PRACTICAL. Little -r, . , , , . understood. By many it is regarded as opposed to theoreti cal. It has become a pert question of our day, its popular " Whether such a branch of knowledge is prac tical ?" " If any practical good arises from pur- Questions relfi tin * to suing such a study?" "If it be not full time gtu diesand that old tomes be permitted to remain untouched in the alcoves of the library, and the minds of the young fed with the more stimulating food of modern progress ?" 352. Such inquiries are not to be answered inquiries: by a taunt. They must be met as grave ques- H ^ to be tions, and considered and discussed with calm ness. They have possession of the public mind ; ^^ influence. they affect the foundations of education ; they 326 UTILITY OF MATHEMATICS. [BOOK III. Their influence and direct the first steps ; they control the very elements from whici systems of public instruction. the very elements from which must spring the Practical: 353. The term "practical," in its common common acceptation, that is, in the sense in which it is often used, refers to the acquisition of useful knowledge by a short process. It implies a sub- whatu stitution of natural sagacity and "mother wit" lmphe8 for the results of hard study and laborious effort. It implies the use of knowledge before its acqui sition ; the substitution of the results of mere experiment for the deductions of science, and the placing of empiricism above philosophy. in this sense, 354. In this view, the practical is adverse to soun ^ learning, and directly opposed to real progress. If adopted, as a basis of national edu cation, it would shackle the mind with the iron fetters of mere routine, and chain it down to the drudgery of unimproving labor. Under such a system, the people would become imita- tors and rule-men. Great and original principles would be lost sight of, and the spirit of inves tigation and inquiry would find no field for its legitimate exercise. Eight But give to "practical" its true and right signification, and it becomes a word of the CHAP. III.] ILLUSTRATED. 327 choicest import. In its right sense, it is the best Best means means of making the true ideal the actual ; that ^^5^ is, the best means of carrying into the business and practical affairs of life the conceptions and deductions of science. All that is truly great in the practical, is but the actual of an antece dent ideal. 355. It is under this view that we now pro- Mathemati cal science : pose to consider the practical advantages of mathematical science. In the two preceding chapters we have pointed out its value as a means of mental development, and as affording facilities for the acquisition of knowledge. We shall now show how intimately it is blended its practical with the every-day affairs of life, and point out some of the agencies which it exerts in giving practical development to the conceptions of the mind. 356. We begin with Arithmetic, as this Arithmetic branch of mathematics enters more or less into all the others. And what shall we say of its practical utility ? It is at once an evidence and element of civilization. By its aid the child in the nursery numbers his toys, the housewife keeps her daily accounts, and the merchant sums up his daily business. The ten little characters, 328 UTILITY OF MATHEMATICS. [BOOK III. which we call figures, thus perform a very im- what figures portant part in human affairs. They are sleepless sentinels watching over all the transactions of trade and commerce, and making known their final results. They superintend the entire busi- Their value, ness affairs of the world. Their daily records exhibit the results on the stock exchange, and of enterprises reaching over distant seas. The used by the mechanic and artisan express the final results of all their calculations in figures. The dimensions in bunding, of buildings, their length, breadth, and height, as well as the proportions of their several parts, are all expressed by figures before the foundation Aid science, stones are laid ; and indeed, all the results of science are reduced to figures before they can be made available in practice. 357. The rules and practice of all the me chanic arts are but applications of mathematical Mathematics science. The mason computes the quantity of useful in the mechanic his materials by the principles of Geometry and the rules of Arithmetic. The carpenter frames his building, and adjusts all its parts, each to the others, by the rules of practical Geometry. Examples. The millwright computes the pressure of the water, and adjusts the driving to the driven wheel, by rules evolved from the formulas of analvsis. CHAP. III.] ILLUSTRATED. 329 358. Workshops and factories afford marked workshops illustrations of the utility and value of practical exhi Wtap . science. Here the most difficult problems are potions of science. resolved, and the power of mind over matter exhibited in the most striking light. To the uninstructed eye of a casual observer, confusion appears to reign triumphant. But all the parts Parts ad justed on a of that complicated machinery are adjusted to general plan. each other, and were indeed so arranged, and according to a general plan, before a single wheel was formed by the hand of the forger. The power necessary to do the entire work was Power first carefully calculated, arid then distributed an d throughout the ramifications of the machinery. Each part was so arranged as to fulfil its office. Every circumference, and band, and cog, has its specific duty assigned it. The parts are Parts fit in their proper made at different places, after patterns formed places, by the rules of science, and when brought to gether, fit exactly. They are but formed parts of an entire whole, over which, at the source of power, an ingenious contrivance, called the Governor, presides. His function is to regulate Governor: the force which shall drive the whole according to a uniform speed. He is so intelligent, and of such delicate sensibility, that on the slightest its functions. increase of velocity, he diminishes the force, and adds additional power the moment the speed 330 UTILITY OF MATHEMATICS. [BOOK III. AII is but slackens. All this is the result of mathematical the result of . TTTI science, calculation. When the curious shall visit these exhibitions of ingenuity and skill, let them not suppose that they are the results of chance and experiment. They are the embodiments, by in telligent labor, of the most difficult investigations of mathematical science. 359. Another striking example of the appli cation of the principles of science is found in steamship: the steamship. in the first place, the formation of her hull, HOW the huii so as to divide the waters with the least resist- ied ance, and at the same time receive from them the greatest pressure as they close behind her, Her masts: is not an easy problem. Her masts are all How to be set at the proper angle, and her sails so adjusted. ac ij uste( j as to gain a maximum force. But the complication of her machinery, unless seen through the medium of science, baffles investi gation, and exhibits a startling miracle. The burning furnace, the immense boilers, the mass- Machinery: i yc cylinders, the huge levers, the pipes, the lifting and closing valves, and all the nicely- adjusted apparatus, appear too intricate to be comprehended by the mind at a single glance. The whole Yet in all this complication in all this variety of principle and workmanship, science has ex- CHAP. III.] ILLUSTRATED. 331 erted its power. There is not a cylinder, whose according to the principles dimensions were not measured not a lever, O f science: whose power was not calculated nor a valve, which does not open and shut at the appointed moment. There is not, in all this structure, a From a . general plan. bolt, or screw r , or rod, which was not provided for before the great shaft was forged, and which does not bear to that shaft its proper proportion. And when the workmanship is put to the test, B>- what means and the power of steam is urging the vessel on lia vigated: her distant voyage, science alone can direct her way. In the captain s cabin are carefully laid away, for daily use, maps and charts of the port which Her charts: he leaves, of the ocean he traverses, and of the coasts and harbors to which he directs his way. On these are marked the results of much scien- Their tific labor. The shoals, the channels, the points uses. of danger and the places of security, are all in dicated. Near by, hangs the barometer, con- Barometer: structed from the most abstruse mathematical formulas, to indicate changes in the weight of the atmosphere, and admonish him of the ap proaching tempest. On his table lie the sextant, sextant: and the tables of Bowditch. These enable him, by observations on the heavenly bodies, to mark his exact place on the chart, and learn his posi- Their uses. tion on the surface of the earth. Thus, practical 332 UTILITY OF MATHEMATICS. [BOOK III. science science, which shaped the keel of the ship to guides the ship : its proper form, and guided the hand of the me chanic in every workshop, is, under Providence, the means of conducting her in safety over the ocean. It is, indeed, the cloud by day and the What pillar of fire by night. Guiding the bark of thusaccom- r pushes, commerce over trackless waters, it brings dis tant lands into proximity, and into political and social relations. " We have before us an anecdote communi- Illustration. cated to us by a naval officer,* distinguished for the extent and variety of his attainments, which shows how impressive such results may Ca t Hairs b ecom 6 in practice. He sailed from San Bias, voyage. on ^ west coas t o f Mexico, and after a voyage its length: of eight thousand miles, occupying eighty-nine days, arrived off Rio de Janeiro ; having in this interval passed through the Pacific Ocean, round- and ed Cape Horn, and crossed the South Atlantic, incidents. without making any land, or even seeing a single sail, with the exception of an American whaler off Cape Horn. Arrived within a week s sail of Rio, he set seriously about determining, by observations lunar observations, the precise line of the ship s taken. course, and its situation in it, at a determinate moment; and having ascertained this within * Captain Basil Hall. CHAP. III.] ILLUSTRATED. 333 from five to ten miles, ran the rest of the way Remarkable by those more ready and compendious methods, known to navigators, which can be safely em ployed for short trips between one known point and another, but which cannot be trusted in long short methods. voyages, where the moon is the only sure guide. " The rest of the tale, we are enabled, by his kindness, to state in his own words : We steered Particulars towards Rio de Janeiro for some days after ta king the lunars above described, and having arrived within fifteen or twenty miles of the Arrival at coast, I hove-to at four in the morning, till the day should break, and then bore up : for although it was very hazy, we could see before us a couple of miles or so. About eight o clock it became so foggy, that I did not like to stand in further, and was just bringing the ship to the wind again, be fore sending the people to breakfast, when it sud denly cleared off, and I had the satisfaction of Discovery of Harbor. seeing the great Sugar-Loaf Rock, which stands on one side of the harbor s mouth, so nearly right ahead that we had not to alter our course above a point in order to hit the entrance of Rio. This was the first land we had seen for three months, First land in , , . i , three alter crossing so many seas, and being set back- mont hg. wards and forwards by innumerable currents and foul winds. The effect on all on board Effect might well be conceived to have been electric ; 334 UTILITY OF MATHEMATICS. [BOOK III. on the crew, and it is needless to remark how essentially the authority of a commanding officer over his crew may be strengthened by the occurrence of such incidents, indicative of a degree of knowledge and consequent power beyond their reach."* Surveying. 360. A useful application of mathematical science is found in the laying out and measure- Measure- ment of land. The necessity of such measure ment of land. ment, and of dividing the surface of the earth into portions, gave rise to the science of Geom- ownership: etry. The ownership of land could not be de- How termined without some means of running boun determined. dary lines, and ascertaining limits. Levelling is also connected with this branch of practical mathematics. By the aid of these two branches of practical science, we measure and determine the area or contents of contents of ground ; make maps of its surface ; measure the heights of hills and mountains ; Rivers, find the directions of rivers ; measure their vol umes, and ascertain the rapidity of their cur rents. So certain and exact are the results, that entire countries are divided into tracts of con venient size, and the rights of ownership fully Certainty, secured. The rules for mapping, and the con- * Sir John Herschel, on the study of Natural Philosophy. CHAP. III.] ILLUSTRATED. 335 ventional methods of representing the surface Mapping, of ground, the courses of rivers, and the heights of mountains, are so well defined, that the nat ural features of a country may be all indicated Features of on paper. Thus, the topographical features of all the known parts of the earth may be cor- Their repre- rectly and vividly impressed on the mind, by a map, drawn according to the rules of art, by the human hand. 361. Our own age has been marked by a Railways, striking application of science, in the construc tion of railways. Let us contemplate for a mo- The problem ment the elements of the problem which is pre- pres sented in the enterprise of constructing a railroad between two given points. In the first place, the route must be carefully Examination examined to ascertain its general practicability, routes!" The surveyor, with his instruments, then ascer- surveys. tains all the levels and grades. The engineer examines these results to determine whether the office of the power of steam, in connection with the best combination of machinery, will enable him to overcome the elevations and descend the decliv ities in safety. He then calculates the curves calculations of the road, the excavations and fillings, the cost of the bridges and the tunnels, if there are any ; and then adjusts the steam-power to meet 336 UTILITY OF MATHEMATICS. [fiOOK III. Completion the conditions. In a few months after the enter- and use. prise is undertaken, the locomotive, with its long train of passenger and freight cars, rushes over the tract with a superhuman power, and fulfils the office of uniting distant places in commer cial and social relations. The striking But that which is most striking in all this, is f ict the fact, that before a stump is grubbed, or a spade put into the ground, the entire plan of the work, having been subjected to careful analysis, is fully developed in all its parts. The construc- The whole tion is but the actual of that perfect ideal which ^bnflB, f the mind forms within itself, and which can spring only from the far-reaching and immuta ble principles of abstract science. 362. Among the most useful applications of practical science, in the present century, is the croton introduction of the Croton water into the city .Deduct. In the Highlands of the Hudson, about fifty miles from the city, the gushing springs of the sources of mountains indicate the sources of the Croton river, which enters the Hudson a few miles below Peekskill. At a short distance from the Principal mouth, a dam fifty-five feet in height is thrown reservoir. across the river, creating an artificial lake for the permanent supply of water. The area of this CHAP, in.] ILLUSTRATED. 337 Their heights. Streams crossed. lake is equal to about four hundred acres. The its area. aqueduct commences at the Croton dam, on a Aqueduct, line forty feet above the level of the Hudson river, and runs, as near as the nature of the ground will permit, along the east bank, till it reaches its final destination in the reservoirs of the city. There are on the line sixteen tun- its tunnels; nels, varying in length from 160 to 1,263 feet, making an aggregate length of 6,841 feet. The heights of the ridges above the grade level of the tunnels range from 25 to 75 feet. Twenty-five streams are crossed by the aqueduct in West- chester county, varying from 12 to 70 feet below the grade line, and from 25 to 83 feet below the top covering of the aqueduct. The Harlem Harlem river: river is passed at an elevation of 120 feet above the surface of the water. The average dimen sions of the interior of the aqueduct, are about seven feet in width and eight feet in height. The width of the Harlem river, at the point its width, where the aqueduct crosses it, is six hundred and twenty feet, and the general plan of the bridge is as follows : There are eight arches, Bridge : each of 80 feet span, and seven smaller arches, each of 50 feet span, the whole resting on piers and abutments. The length of the bridge is its length: 1,450 feet. The height of the river piers from the lowest foundation is 96 feet. The arches 22 338 UTILITY OF MATHEMATICS. [BOOK III. its height: are semi-circular, and the height from the low est foundation of the piers to the top of the its width, parapet is 149 feet. The width across, on the top, is 21 feet. To afford a constant supply of water for dis tribution in the city two large reservoirs have Receiving been constructed, called the receiving reservoir and the distributing reservoir. The surface of the receiving reservoir, at the water-line, is equal cts extent, to thirty-one acres. It is divided into two parts by a wall running east and west. The depth of Dc>pth of water in the northern part is twenty feet, and water. . in the southern part thirty feet. Distributing The distributing reservoir is located on the : highest ground which adjoins the city, known its capacity, as Murray Hill. The capacity of this reservoir is equal to 20,000,000 of gallons, which is about one-seventh that of the receiving reservoir, and the depth of water is thirty- six feet. Power The full power of science has not yet been illustrated. A perfect plan of this majestic structure was arranged, or should have been, before a stone was shaped, or a pickaxe put into the ground. The complete conception, by a single mind, of its general plan and minutest details, was necessary to its successful prosecu- whatitac- t j on> j t was w ithin the range and power of complished. science to have given the form and dimensions CHAP. III.] ILLUSTRATED. 339 of every stone, so that each could have been shaped at the quarry. The parts are so con- nected by the laws of the geometrical forms, that the dimensions and shape of each stone was exactly determined by the nature of that portion of the structure to which it belonged. 363. We have presented this outline of the view of the Croton aqueduct mainly for the purpose of aqueduct: illustrating the power and celebrating the tri- why given. umphs of mathematical science. High intel lect, it is true, can alone use the means in a work so complicated, and embracing so great a variety of intricate details. But genius, even Little ao _ of the highest order, could not accomplish, with- ^f out continued trial and laborious experiment, science - such an undertaking, unless strengthened and guided by the immutable truths of mathematical science. 364. The examination of this work cannot what . ,, , , . , . c science has but fill the mind with a proud consciousness of done the power and skill of man. The struggling brooks of the mountains are collected together accumulated conducted for forty miles through a subterranean channel, to form small lakes in the vicinity of a populous city. From these sources, by an unseen process, the 340 UTILITY OF MATHEMATICS. [BOOK III. pure water is carried to every dwelling in the large metropolis. The turning of a faucet de- conse- livers it from a spring at the distance of fifty quences which have miles, as pure as when it gushes from its granite followed. I MI mi r i i hills. Inat unseen power 01 pressure, which resides in the fluid as an organic law, exerts its force with unceasing and untiring energy. To minds enlightened by science, and skill directed by its rules, we are indebted for one of the no blest works of the present century. May we conclusion, not, therefore, conclude that science is the only sure means of giving practical development to those great conceptions which confer lasting benefits on mankind ? " All that is truly great in the practical, is but the result of an antece dent ideal." APPENDIX. A COURSE OF MATHEMATICS WHAT IT SHOULD BE. 365. A COURSE of mathematics should pre- A course sent the outlines of the science, so arranged, ex- Mathematics, plained, and illustrated as to indicate all those general methods of application, which render it effective and useful. This can best be done by a series of works embracing all the topics, and in which each topic is separately treated. 366. Such a series should be formed in ac- HOW it cordance with a fixed plan ; should adopt and ^"med. 6 use the same terms in all the branches ; should be written throughout in the same style ; and present that entire unity which belongs to the unity of the subject itself. Bubject> 367. The reasonings of mathematics and Reasonings the processes of investigation, are the same in 342 APPENDIX. the same in every branch, and have to be learned but once, if the same system be studied throughout. The Different different kinds of notation, though somewhat un- kinds of no tation, like in the different subjects of the science, are, in fact, but dialects of a common language. Language 359 if tneri} the language is, or may be need be learned but made essentially the same in all the branches of mathematical science ; and if there is, as has been fully shown, no difference in the processes in what f reasoning, wherein lies that difficulty in the difficult"? 6 acc l u ^ s ^i on f mathematical knowledge which is often experienced by students, and whence the origin of that opinion that the subject itself is dry and difficult ? A 369. Just in proportion as a branch of know- general law, if known, ledge is compactly united by a common law, is abject jN0r. tne f acm ty of acquiring that knowledge, if we observe the law, and the difficulty of acquiring Faculties it, if we pay no attention to the law. The study mauwmatics. ^ mathematics demands, at every step, close attention, nice discrimination, and certain judg ment. These faculties can only be developed HOW first by culture. They must, like other faculties, pass cultivated: . through the states ot miancy, growth, and ma turity. They must be first exercised on sensible and simple objects ; then on elementary ab- APPENDIX. 343 stract ideas ; and finally, on generalizations and on what ,, , 1-1 finally exer- the higher combinations of thought in the pure cised ideal. 370. Have educators fully realized that the Arithmetic the most im- first lessons in numbers impress the first elements P0 rtant of mathematical science ? that the first con nections of thought which are there formed be come the first threads of that intellectual warp which gives tone and strength to the mind ? Have they yet realized that every process is, or AH the should be, like the stone of an arch, formed to nected< fill, in the entire structure, the exact place for which it is designed ? and that the unity, beauty, and strength of the whole depend on the adapta tion of the parts to each other ? Have they sufficiently reflected on the confusion which must Necessity of unity in all arise from attempting to put together and nar- the parts, monize different parts of discordant systems ? to blend portions that are fragmentary, and to unite into a placid and tranquil stream trains of thought which have not a common source ? 371. Some have supposed that Arithmetic may be well taught and learned without the aid of a text-book ; or, if studied from a book, that A text-book the teacher may advantageously substitute his own methods for those of the author, inasmuch 344 APPENDIX. tobefoi- as such substitution is calculated to widen the field of investigation, and excite the mind of the pupil to new inquiries. Reasons. Admitting that every teacher of reasonable intelligence, will discover methods of communi cating instruction better adapted to the peculiar ities of his own mind, than all the methods em- Evenabet- ployed by the author he may use; will it be safe, ter method, when substi- as a general rule, to substitute extemporaneous noth armo- methods for those which have been subjected to ^ e analysis of science and the tests of expe- of the work. r j ence ? J s it sa f e to substitute the results of known laws for conjectural judgments ? But if they are as good, or better even, as isolated pro cesses, will they answer as well, in their new places and connections, as the parts rejected ? illustration. Will the balance- wheel of a chronometer give as steady a motion to a common watch as the more simple and less perfect contrivance to which all the other parts are adapted ? 372. If these questions have significance, we one of the have found at least one of the causes that have reasons why ..,._.. mathematics impeded the advancement 01 mathematical sci ence, viz. the attempt to unite in the same course of instruction fragments of different systems ; thus presenting to the mind of the learner the same terms differently defined, and the same APPENDIX. 345 principles differently explained, illustrated, and applied. It is mutual relation and connection connection very impor- which bring sets of facts under general laws ; it tant. is mutual relation and connection of ideas which form a process of science ; it is the mutual con nection and relation of such processes which constitute science itself. 373. I would by no means be understood as A teacher expressing the opinion that a student or teacher should rei * i many books, of mathematics should limit his researches to a and teach one system. single author ; for, he must necessarily read arid study many. I speak of the pupil alone, who must be taught one method at a time, and taught that well, before he is able to compare different methods with each other. ORDER OF THE SUBJECTS ARITHMETIC. 374. Arithmetic is the most useful and Arithmetic: simple branch of mathematical science, and is the first to be taught. If, however, the pupil has time for a full course, I would by no means connection recommend him to finish his Arithmetic before A igebra. studying a portion of Algebra. 346 APPENDIX. ALGEBRA. Algebra: ^5. Algebra is but a universal Arithmetic, with a more comprehensive notation. Its ele ments are acquired more readily than the higher and hidden properties of numbers ; and indeed, the elements of any branch of mathematics are more simple than the higher principles of the HOW preceding subject ; so that all the subjects can it should be . . studied: best be studied in connection with those which precede and follow. should 376. Algebra, in a regular course of instruc- tion, should precede Geometry, because the ele mentary processes do not require, in so high a why. degree, the exercise of the faculties of abstrac tion and generalization. But when we have when completed the equation of the second degree, rijouMbe the processes become more difficult, the abstrac- commenced. t j ons more perfect, and the generalizations more extended. Here then I would pause and com mence Geometry. GEOMETRY. Geometry. 377. Geometry, as one of the subjects of mathematical science/has been fully considered in Book II. It is referred to here merely to mark its place in a regular course of instruction. APPENDIX. 347 TRIGONOMETRY PLANE AND SPHERICAL. 378. The next subject in order, after Geom- try : etry, is Trigonometry : a mere application of the principles of Arithmetic, Algebra, and Geometry what it is. to the determination of the sides and angles of triangles. As triangles are of two kinds, viz. those formed by straight lines and those formed by the arcs of great circles on the surface of a sphere ; so Trigonometry is divided into two TWO kinds, parts : Plane and Spherical. Plane Trigonom etry explains the methods, and lays down the plane - necessary rules for finding the remaining sides and angles of a plane triangle, when a sufficient number are known or given. Spherical Trigo- spherical, nometry explains like processes, and lays down similar rules for spherical triangles. SURVEYING AND LEVELLING. 379. The application of the principles of Trigonometry to the measurement of portions of the earth s surface, is called Surveying; and surveying. similar applications of the same principles to the determination of the difference between the dis tances of any two points from the centre of the earth, is called Levelling. These subjects, which Levelling, follow Trigonometry, not only embrace the va- 348 APPENDIX. what they rious methods of calculation, but also a descrip tion of the necessary Instruments and Tables. They should be studied immediately after Trigo nometry ; of which, indeed, they are but appli cations. DESCRIPTIVE GEOMETRY. Descriptive 380. Descriptive Geometry is that branch Geometry : of mathematics which considers the positions of the geometrical magnitudes, as they may exist in space, and determines these positions by refer ring the magnitudes to two planes called the Planes of Projection. its nature. It is, indeed, but a development of those gen eral methods, by which lines, surfaces, and solids may be presented to the mind by means of drawings made upon paper. The processes of what its this development require the constant exercise of StU pTishe C s ra ~ tne conceptive faculty. All geometrical mag nitudes may be referred to two planes of pro jection, and their representations on these planes will express to the mind, their forms, extent, and also their positions or places in space. From HOW. these representations, the mind perceives, as it were, at a single view, the magnitudes them selves, as they exist in space ; traces their boun daries, measures their extent, and sees all their parts separately and in their connection. APPENDIX. 349 In France, Descriptive Geometry is an impor- HOW , , regarded in tant element of education. It is taught in most Frauce> of the public schools, and is regarded as indis pensable to the architect and engineer. It is, indeed, the only means of so reducing to paper, and presenting at a single view, all the compli cated parts of a structure, that the drawing or representation of it can be read at a glance, and all the parts be at once referred to their appropri ate places. It is to the engineer or architect not its value only a general language by which he can record branch. and express to others all his conceptions, but is also the most powerful means of extending those conceptions, and subjecting them to the laws of exact science. SHADES, SHADOWS, AND PERSPECTIVE. 381. The application of Descriptive Geom etry to the determination of shades and shadows, shades, Shadows, as they are found to exist on the surfaces of and bodies, is one of the most striking and useful ap- pers P ectlve - plications of science ; and when it is further extended to the subject of Perspective, we have all that is necessary to the exact representation of objects as they appear in nature. An accu rate perspective and the right distribution of light and shade are the basis of every work of 3"0 APPENDIX mm the fine arts. Without them, the sculptor ar.d the painter would labor in vain : the chisel of Canova would give no life to the marble, nor the touches of Raphael to the canvas. ANALYTICAL GEOMET1Y. f 382. Analytical Geometry is the next sub ject in a regular course of mathematical study. though it may be studied before Descriptive Ge ometry. The importance of this subject cannut be exaggerated. In Algebra, the symbols of quantity have generally so close a connection with numbers, that the mind scarcely realizes the extent of the generalization : and the power of analysis, arising from the changes that mav take place among the quantities which the sym bols represent, cannot be fully explained and de- But in Analytical Geometry, where all the magnitudes axe brought under the power of anal- jmm, and all their properties developed by the combined processes of Algebra and Geometry, we are brought to feel the extent and potency of those methods which combine in a single equa tion every discovered and undiscovered property of every fine, straight or curved, which can be tin ii Mil by the intersection of a cone and plane. APPENDIX. 3". To develop every property of the Conic Sec- itsextoi. tions from a single equation, and that an equa tion only of the second degree, by the known processes of Algebra, and thus interpret the re sults, is a far different exercise of the mind from that which arises from searching them out by the tedious and disconnected methods of separate propositions. The first traces all from an inex- i methods haustible fountain by the known laws of analyti cal investigation, applicable to all- similar cases, while the latter adopts particular processes ap plicable to special cases only, without any gen eral law of connection. DIFFERENTIAL AND INTEGRAL CALCULUS. 383. The Differential and Integral Calculus presents a new view of the power, extent, and applications of mathematical science. It should be carefully studied by all who seek to make high attainments in mathematical knowledge, or who desire to read the best works on Natural and Experimental Philosophy. It is that field of mathematical investigation, where genius may exert its highest powers and find its most certain rewards. INDEX. ABSTRACTION That faculty of the mind which enables us, in contem plating any object to attend exclusively to some par ticular circumstance, and quite withhold our attention from the rest, Section 12. " Is used in three senses, 13. Abstract Quantity, 75, 96. Addition, Readings in, 116. Examples in, 151. " of Fractions, Rule for, 191. " Combinations in, 192, 193. Definitions of, 203. " One principle governs all operations in, 232. ^Etna, How far designated by the term mountain, 20. A Geometrical Proportion, 168. ALGEBRA A species of Universal Arithmetic, in which letters and signs are employed to abridge and generalize all pro cesses involving numbers, 280. " Divided into two parts, 280. " Difficulties of, from what arising, 286. " Principles of, deduced from definitions and axioms, 297. " Should precede Geometry in instruction, 376. Alphabet of the language of numbers, 80, 113, 114. " Language of Arithmetic, formed from, 192. Analytical Form, for what best suited, 71, 89. ANALYSIS A term embracing all the operations that can be performed on quantities represented by letters, 87, 88, 274, 327. " It also denotes the process of separating a complex whole into its parts, 89. 44 of problems in Arithmetic, 175, 176. 23 354 ANALYSIS. INDEX. Angles.. Apothecaries APPREHENSION. AREA or .. Argument Arguments, Aristotle ARISTOTLE S .Three branches of, Sections 279, 285, 286. First notions of, how acquired, 317. Problems it, has solved, 344-347. .Right angle, the unit of, 250. A class of Geometrical Magnitudes, 273. Weight Its units and scale, 138. ..Simple apprehension is the notion (or conception) of an object in the mind, 7. Incomplex apprehension is of one object or of several without any relation being perceived between them, 7. Complex is of several with such a relation, 7. .CONTENTS, Number of times a surface contains its unit of measure, 141. with one premiss suppressed is called an Enthymeme, 47. Two kinds of objections to an, 47. Every valid, may be reduced to a syllogism, 52. at full length, a syllogism, 56. concerned with connection between premises and conclu sion, 57. Where the fault (if any) lies, 69. In reasoning we make use of, 42. Examples of unsound, 50. Rules for examining, 70. did not mean that arguments should always be stated syllogistically, 53. accused of darkening his demonstrations by the use of symbols, 57 His philosophy not progressive, 334. DICTUM Whatever is predicated (that is, affirmed or de nied) universally, of any class of things, may be predicated, in like manner (viz. affirmed or denied), of any thing comprehended in that class, 54. " Keystone of his logical system, 54. " Objections to, 54, 55. a generalized statement of all demonstration, 55. " applied to terms represented by letters, 56. " not complied with, 59, 60. " All sound arguments can be reduced to the form to which it applies, 65, 66. INDEX. 355 ARITHMETIC. Arithmetical ART .Is both a science and an art, Section 172. It is a science in all that relates to the properties, laws, and proportions of numbers, 172. It is an art in all that concerns their application, 173. Processes of, not affected by the nature of the objects, 43. Illustration from, 45. How its principles should be explained, 174 Its requisitions as an art, 177. Faculties cultivated by it, 180. Application of principles, 188. Generally preceded by a smaller treatise, 190, Methods of placing subjects examined, 191. Combinations in, 192-199. What its study should accomplish, 206. Art of, its importance, 206. Elementary ideas of, learned by sensible objects, 207. Principles of, how they should be taught, 208. FIRST, what it should accomplish, 214. " arrangement of lessons, 214-223. " what should be taught hi it, 226. SECOND, should be complete and practical, 227. " arrangement of subjects, 228. " introduction of subjects, 229. " reading of figures should be constantly prac tised, 230. THIRD, the subject now taught as a science, 231. " requirements from the pupil for, 231. " Reduction and the ground rules brought under one principle, 232. " design of, methods must differ from smaller works, 233. " examples in the ground rules, 234. " what subjects should be transferred from elemen tary works, 235. Practical utility of, 356, 357. should not be finished before Algebra is commenced, 374. Proportion, 163. Ratio, 163. .The application of knowledge to practice, 22. Its relations to science, 22. 356 INDEX. ART. A single one often formed from several sciences, Section 2 2. of Arithmetic, 173, 177, 182. Astronomy brought by Newton within the laws of mechanics, 337. How it became deductive, 339. Mathematics necessary in, 341. Authors, methods of finding ratio, 165, 170. " of placing Rule of Three, 187. quotations from, on Arithmetic, 201-204 definition of proportion, 268. Auxiliary Quantities, 259, 261. Avoirdupois Weight, its units and scale, 136 AXIOM A self-evident truth, 27, 97. Axioms of Geometry, process of learning them, 27. or canons, for testing the validity of syllogisms, 67. of Geometry established by Induction, 73. for forming numbers, 79. for comparison relate to equality and inequality, 102. " for inferring equality, 102, 258, 260, 264. " " " inequality, 102. employed in solving equations, 278, 311. Bacon, Lord, Quotation from, 328. Foundation of his Philosophy, 334 ; its subject Nature, 335, page 12. " His system inductive, 334. " Object and means of his philosophy, 335 Barometer, Construction and use of, 359 Barrow, Dr., Quotation from, 328, 340. Belief essential to knowledge, 23. " and disbelief are expressed in propositions, 36. Blakewell, steps of his discovery, 32. Bowditch, Tables of, used in Navigation, 359. BREADTH A dimension of space, 82. Bridge, Harlem, description of, 362. CALCULUS, In its general sense, means any operation performed on algebraic quantities, 281, 282. " Differential and Integral, 283-285, 383. Canons for testing the validity of syllogisms, 67. Cause and effect, their relation the scientific basis of induction, 33. INDEX. 357 Chemist, Illustration, Section 53 ; idea of iron, 322. Chemistry aided by Mathematics, 342. CIRCLE A portion of a plane included within a curve, all the points of which are equally distant from a certain point within called the centre, 244. " The only curve of Elementary Geometry, 244. " Property of, 256. Circular Measure, its units and scale, 149. CLASSES Divisions of species or subspecies, in which the charac teristic is less extensive, but more full and complete, 16. CLASSIFICATION..... The arrangement of objects into classes, with reference to some common and distinguishing characteristic, 16. " Basis of, may be chosen arbitrarily, 20. Coefficient of a letter, 291 ; of a product, 292. Differential, 283, 284. Coins should be exhibited to give ideas of numbers, 133. Combinations in Arithmetic, 192-199. " taught in First Arithmetic, 216-218. Comets, Problem with reference to, 347. Comparison, Knowledge gained by, 95. " Reasoning carried on by, 25, 307. CONCLUSION The third proposition of a syllogism, 40. " in Induction, broader than the premises, 31. " deduced from the premises, 40, 41, 46, 47, 49. " contradicts a known truth, in negative demonstrations, 264, 265. Concrete Quantity, 75, 96. Conjunctions causal, illative, 48. " denote cause and effect, premiss and conclusion, 48. CONSTANTS Quantities which preserve a fixed value throughout the same discussion or investigation, 282, 283, 313. " represented by the first letters of the alphabet, 284 COPULA That part of a proposition which indicates the act of judgment, 38. " must be "is" or "is not," 38, 39. Cousin, quotation from, 180. Curves, circumference of circle the simplest of, 239. Croton river, its sources, 362. " dam, its construction, 362 ; lake, area of, 362 aqueduct, description of, 362. 358 INDEX. Decimals, language and scale for, Sections 156, 157. DEDUCTION A process of reasoning by which a particular truth is in ferred from other truths which are known or admitted, 34. " Its formula the syllogism, 34. Deductive Sciences, why they exist, 98. " " aid they give in Induction, 335. DEFINITION A metaphorical word, which literally signifies laying down a boundary, 1. " Is of two kinds, 1. " Its various attributes, 2-5. Definitions, General method of framing, 3. " Rules for framing, 5 (Note). " and axioms, tests of truth, 97, 99. " signs of elementary ideas, 200. " Necessity of exact, 200. DEMONSTEATION ...A series of logical arguments brought to a conclusion, in which the major premises are definitions, axioms, or propositions already established, 237. " of a demonstration, 55. " to what applicable, 238. " of Proposition I. of Legendre, 258. " positive and negative, 262-265. " produces the most certain knowledge, 326. Descartes, originator of Analytical Geometry, 281 Dictum, Aristotle s, 54, 55, 66. DIFFERENTIAL AND INTEGRAL CALCULUS. The science which notes the changes that take place according to fixed laws estab lished by algebraic formulas, when those changes ;ir<: indicated by certain marks drawn from the variable symbols, 283. Coefficients Marks drawn from the variable symbols, 283, 284. and Integral Calculus Difference between it and Ana lytical Geometry, 284. " " " " "What persons should study it, 383. Discussion of an Equation, 308. DISTRIBUTION A term is distributed, when it stands for all its signifi- cates, 61. g A term is not distributed when it stands for only a part of its significates, 61. INDEX. 359 Distribution, Words which mark, not always expressed, Section 62. Division, Readings in, 123 ; examples in, 154. " Combinations in, 196, " All operations in, governed by one principle, 232. " of quantities, how indicated, 294. Dry Measure, Its units and scale, 147. Duodecimal units, 142-144. English Money, Its units and scale, 135. ENTHYMEME An argument with one premiss suppressed, 47. EQUAL. Two geometrical figures are said to be equal when they can be so applied to each other as to coincide through out their whole extent, 255, 312. EQUALITY In Geometry expresses that two figures coincide. In Algebra it merely implies that each member of an equation contains the same unit an equal number of times, 312. EQUATION An analytical formula for expressing equality, 307-312. " A proposition expressed algebraically, in which equality is predicated of one quantity as compared with an other, 309. " either abstract or concrete, 310. Equations, subject of, divided into two parts, 308. Five axioms for solving, 311. EQUIVALENT Two geometrical figures are said to be equivalent when they contain the same unit of measure an equal num ber of times, 255. Examples in ground rules of Third Arithmetic, 234. Of little use to vary forms of, without changing the prin ciples of construction, 236. Experiment, in what sense used, 25 (Note). EXPONENT An expression to show how many equal factors are em ployed, 293. Extremes. Subject and predicate of a proposition, 38, 67. FACT Any thing which has been or is, 24. " Knowledge of, how derived, 25. " In what sense used, 25. " regarded as a genus, 25. Factories, value of science in, 358. 360 INDEX FALLACY Any unsound mode of arguing which appears to demand our conviction, and to be decisive of the question in hand, when in fairness it is not, Section 68. Illustration of, 53. " Example and analysis of, 59, 60. " Material and Logical, 69. " Rules for detecting, 70. Federal Money, units increase by scale of tens, 129, 134. " Methods of reading, 129, 134. FIGURE A portion of space limited by boundaries, 83. " Each geometrical, stands for a class, 277. Figures in Arithmetic show how many times a unit is taken, 125. do not indicate the kind of unit, 125. Laws of the places of, 126, 127. " have no value, 128, 201. " Methods of reading, 130 ; of writing, 199. " Definitions of, 201, 202. " should be early used in Arithmetic, 219. First Arithmetic, what should be taught in it, 226. " Faculties to be cultivated by it, 214. " Construction of the lessons, 214-218. " Lesson in Fractions, 220-224. " Tables of Denominate Numbers Examples, 225. Fractions come from the unit one, 132. " should be constantly compared with one, 162. " Reasons for placing Common Fractions immediately after Division examined, 189. " not " unexecuted divisions," 189. " Elementary idea of, 189. " Expression for, the same as for Division, 189. Definitions of, 204. " Lessons in, in First Arithmetic, 220-224. FRACTIONAL units, 155 ; orders of, 156 ; language of, 156-159, 197. " " three things necessary to their apprehension, 160. " advantages of, 161. " " two things necessary to their being equal, 161. Galileo, imprisoned in the 17th century, 343. GENERALIZATION.... The process of contemplating the agreement of several objects in certain points, and giving to all and each of INDEX. 36] these objects a name applicable to them in respect to this agreement, Section 14. Generalization implies abstraction, 14. " must be preceded by knowledge, 1 84. GENUS The most extensive term of classification, and conse quently the one involving the fewest particulars, 16, 17. " HIGHEST. That which cannot be referred to a more ex tended classification, 19. " SUBALTERN. A species of a more extended classifica tion, 18. Geometrical Magnitudes, three classes of, 238, 273. " " do not involve matter, 247. " " their boundaries or limits, 247. " " each has its unit of measure, 252. " " analysis of comparison, 270, 271. " " to what the examination of properties has reference, 273. " Proportion, 163 ; Ratio, 163 ; Progression, 170. GEOMETRY Treats of space, and compares portions of space with each other, for the purpose of pointing out their properties and mutual relations, 237. * Why a deductive science, 257. First notions of, how acquired, 318-320. Practical utility of, 357. u Origin of the science, 360. " Its place in a course of instruction, 377. tt ANALYTICAL, Examines the properties, measures, and re lations of the Geometrical Magnitudes by means of the analytical symbols, 281, 282. " originated with Descartes, 281. u " difference between it and Calculus, 284. " " its importance, extent, and methods, 382. " DESCRIPTIVE. That branch of mathematics which con siders the positions of the Geometrical Magnitudes as they may exist in space, and determines these positions by re ferring the magnitudes to two planes called the Planes of Projection, 380. how regarded in France, 380. Governor, functions of, in machinery, 358. 332 INDEX. Grammar Gravitation, defined, Section 113. Law of, 32, 344. Hall, Captain s, voyage from San Bias to Rio Janeiro, 359. Harlem river, Bridge over, and width, 362. Herschel, Sir John, Quotation from, 27, 322, 341, 359. Hull of the steamship, how formed, 359. Illative Conjunctions, 48. ILLICIT PROCESS When a term is distributed in the conclusion which was not distributed in one of the premises, 67. Indefinite Propositions, 62. Index of a root, 295. INDUCTION Is that part of Logic which infers truths from facts, 30-33. Logic of, 30. " supposes necessary observations accurately made, 32. " Example of, Blakewell, 32 ; of Newton, 32. " based upon the relation of cause arid effect, 33. u Reasoning from particulars to generals, 34. " its place in Logic, 72. " how thrown into the form of a syllogism, 74, 99. " Truths of, verified by Deduction, 335, 336. Inertia proportioned to weight, 268. INFINITY, The limit of an increasing quantity, 302-306. Integer Numbers, why easier than fractions, 162. " constructed on a single principle, 231. INTUITION Is strictly applicable only to that mode of contemplation, in which we look at facts, or classes of facts, and im mediately apprehend their relations, 27. Iron, different ideas attached to the word, 322. JUDGMENT Is the comparing together in the mind two of the notions (or ideas) which are the objects of apprehension, and pronouncing that they agree or disagree, 8. " is either Affirmative or Negative, 8. Kant, quotation from, 21. KNOWLEDGE Is a clear and certain conception of that which is true, 23. " facts and truths elements of, 25. of facts, how derived, 25. INDEX. *363 Knowledge some possessed antecedently to reasoning, Section 29. the greater part matter of inference, 29. must precede generalization, 184. two ways of increasing, 323. cannot exceed our ideas, 323. the increase of, renders classification necessary, page 20. LANGUAGE Affords the signs by which the operations of the mind are recorded, expressed, and communicated, 10. " Every branch of knowledge has its own, 11. " of numbers, 80; of mathematics, 91, 92. of mathematics must be thoroughly learned, 92. " " its generality, 93. " for fractional units, 156, 159, 197. Arithmetical, 192-199. " exact, necessary to accurate thought, 205. " of Arithmetic, its uses, 219. of Algebra, the first thing to which the pupil s mind should be directed, 290. " Culture of the mind by the use of exact, 322. Laws of Nature, Science makes them known, 21, 315. " refers individual cases to them, 55. generalized facts, 55, page 14. " include all contingencies, 332. " every diversity the effect of, 346. one dimension of space, 8 1. in First Arithmetic, how arranged, 214. " " " their connections, 218. may stand for all numbers, 276. " represents things in general, 277. LEVELLING The application of the principles of Trigonometry to the determination of the difference between the distances of any two points from the centre of the earth, 379. " Its practical uses, 360. Limit, definition of, 306. LINE One dimension of space, 83, 239. " A straight line does not change its direction, 83, 239, 318. " Curved line, one which changes its direction at every point, 83, 239. Axiom of the straight, 239. Length Lessons Letter 364 INDEX. Lines, limits of, Section 247 " Auxiliary, 259. Liquid Measure, Its units and scale, 146. " Local value of a figure," has no significance, 128, 201. Locke, Quotation from, 323. LOGIC .: Takes note of and decides upon the sufficiency of the evi dence by which truths are established, 29. " Nearly the whole of science and conduct amenable to, 29. " of Induction, its nature, 30. " Archbishop Whateley s views of, 72. " Mr. Mill s views of, 72. Logical Fallacy, 69. Machinery of factories arranged on a general plan, 358. of the steamship, 359. Major Premiss, often suppressed, cannot be denied, 46. " ultimate, of Induction, 74, 99. Major Premises of Geometry, 237, 257. Mansfield, Mr., Quotation from, 325, 327. MARK The evidence contained in the attributes implied in a general name, by which we infer that any thing called by that name possesses another attribute or set of at tributes. For example : " All equilateral triangles are equiangular." Knowing this general proposition, when we consider any object possessing the attributes implied in the term " equilateral triangle," we may infer that it possesses the attributes implied in the term " equian gular;" thus using the first attributes as a mark or evidence of the second. Hence, whatever possesses any mark possesses those attributes of which it is a mark, 98, 257 259. Masts of the steamship, how placed, 359. Material Fallacy, 69. Mathematical Reasoning conforms to logical rules, 73. every truth established by, is developed by a process of Arithmetic, Geometry, or Analy sis, or a combination of them, 90. MATHEMATICS The science of quantity, 76. " Pure, embraces the principles of the science, 76-78. " on what based, 97. INDEX. 365 MATHEMATICS Mixed, embraces the applications, Section 76. " Primary signification, 77. " Language of, 91. * " Exact science," 97. " Logical test of truth in, 97. " a deductive science, 97, 98. u concerned with number and space, 73, 76, 78, 101. " What gives rise to its existence, 100. " Why peculiarly adapted to give clear ideas, 324-326, 329 " a pure science, 329. " considered as furnishing the keys of knowledge, 331. " Widest applications are in nature, 334. Effects on the mind and character, 328, 340. Guidance through Nature, 340. " Its necessity in Astronomy, 341. Results reached by it, 349, 350. * Practical advantages of, 355. " What a course of, should present, and how, 365, 366. Reasonings of, the same in each branch, 367. " Faculties required by, 369. Necessity of, to the philosopher, page 16. MEASURE A term of comparison, 94. Unit of. should be exhibited to give ideas of numbers, 133. " for lines, surfaces, solids, 249. u of a magnitude, how ascertained, 249. Middle Term distributed when the predicate of a negative proposi tion, 64. " When equivocal, 67. Mill, Mr. his views of Logic, 72, 74. Mind, Operations of, in reasoning, 6. Abstraction a faculty, process, and state of, 13. Processes of, which leave no trace, 68. Faculties of, cultivated by Arithmetic, 180. Thinking faculty of, peculiarly cultivated by mathemat ics, 325, 326. Minus sign, Power of, fixed by definition, 297. Motion proportional to force impressed, 268. Multiplication, Readings in, 122 ; examples in, 153. What the definition of, requires, 177. Combinations in, 195. 366 INDEX. Multiplication, All operations in, governed by one principle, Section 232. " in Algebra, illustrations of, 299-301. Names, Definitions are of, 1. " given to portions of space, and defined in Geometry, 238. Naturalist determines the species of an animal from examining a bone, 333. Negative premises, nothing can be inferred from, 67. " demonstration, its nature, 263, 265 ; illustration of, 264. Newton, his method of discovery, 32. " changed Astronomy from an experimental to a deductive science, 337, 339. Non-distribution of terms, 61. " Word " some" which marks, not always expressed, 62 NUMBERS Are expressions for one or more things of the same kind, 79, 106. " How learned, 79. " Axioms for forming, 79, 304. " Three ways of expressing, 107. u Ideas of, complex, 108, 124. " Two things necessary for apprehending clearly, 110. " Simple and Denominate, 112. Examples of reading Simple, 130. " Two ways of forming from ONE, 131. " first learned through the senses, 133, 316. " Two ways of comparing, 163. " compared, must be of the same kind, 171, 175. Definitions of, 201, 202. " must be of something, 275. may stand for all things, 276. " First lessons in, impress the first elements of mathemati cal science, 370. Olmsted s Mechanics, quotation from, 269. Optician, Illustration, 212. Oral Arithmetic, its inefficiency without figures, 219. Order of subjects in Arithmetic, 182, 188. PARALLELOGRAM ...A quadrilateral having its opposite sides taken two and two parallel, 242. INDEX. 367 Parallelogram regarded as a species, Section 17 ; as a genus, 18. " Properties of, 256. Particular proposition, 62. " premises, nothing can be proved from, 67. Pendulum, the standard for measurement, 253. Philosophy, Natural, originally experimental, 337. " has been rendered mathematical, 337. Place idea attached to the word, 81. " designates the unit of a number, 202. PLANE That with which a straight line, having two points in common, and any how placed, will coincide, 240. " First idea of, how impressed, 319. PLANE FIGURE ... .Any portion of a plane bounded by linen, 240. Plane Figures in general, 243. POINT That which has position in space without occupying any part of it, 81. Points, extremities or limits of a line, 239. Practical Rules in Arithmetic, 177, 178. " The true, 207, must be the consequent of science, 228. " Popular meaning of, 351, 353. " Questions with regard to, 351, 352. " Consequences of an erroneous view of, 354. " True signification of, 354. Practice precedes theory, but is improved by it, 42, " without science is empiricism, page 13. PREDICATE That which is affirmed or denied of the subject, 38 " Distribution, 63. Non -distribution, 63. " sometimes coincides with the subject, 63. PREMISS Each of two propositions of a syllogism admitted to be true, 40. MAJOR PREMISS The proposition of a syllogism which contains the predicate of the conclusion, 40. MINOR PREMISS The proposition of a syllogism which contains the subject of the conclusion, 40. Pressure, a law of fluids, 364. Principle of science applied, 22 " on which valid arguments are constructed, 52. " Value of a, greater as it is more simple, 54. Aristotle s Dictum, a general, 55. 368 INDEX. Principle the same in the ground rules for simple and denominate numbers, Sections 151-154, 232. " of science and rule of art, 179. Principles should be separated from applications, 186, 187. of science are general truths, 208. " of Arithmetic, how taught, 208. should precede practice, 229. " of Mathematics, deduced from definitions and axioms, 297 Process of acquiring mathematical knowledge, 316-320. Product of several numbers, 292. Progression, Geometrical, 170. Property of a figure, 256. PROPORTION The relation which one quantity bears to another with re spect to its being greater or less, 163, 267-269 " Arithmetical and Geometrical, 163. Reciprocal or Inverse, 269. " of geometrical figures, 270-273. PROPOSITION A judgment expressed in words, 35. All truth and all error lie in propositions, also answers to all questions, 36. " formed by putting together two names, 37. " consists of three parts, 38. subject, and predicate, called extremes, 38. " Affirmative, 39 ; Negative, 39. Three propositions essential to a syllogism, 40. " A Universal, 62. " Particular, 62. QUADRILATERAL.... A portion of a plane bounded by four straight lines, 242. " * regarded as a genus, 17. " Different varieties of, 242. Quality of a proposition refers to its being affirmative or nega tive, 63. Quantities only of the same kind can be compared, 267. " Two classes of, in Algebra, 287, 313. " " " " in the other branches of Analysis, 282, 283, 313. " compared, must be equal or unequal, 102, 307. QUANTITY Is a general term applicable to everything which can be increased or diminished, and measured, 75, 321. INDEX. 3Gi) Quantity, Abstract, does not involve matter, Sections 75, 96. " Concrete does, 75, 96, " Propositions divided according to, 62. " presented by symbols, 93. " consists of parts which can be numbered, 276 " Constant, 282. " Variable, 282. " Five operations can be performed on, 288, 295. " represented by five signs, 289. " Nature of, not affected by the sign, 290, 296. Questions known, when all propositions are known, 36. ** with regard to number and space, 78. Analysis of, 175, 176. u Difficult, in Fractions avoided, 191 " with regard to methods of instruction, 371. Quotations from Kant, 21 ; Sir John Herschel, 27, 322, 341, 359 ; Cousin, 180; Olmsted s Mechanics, 268; Locke, 323 ; Mansfield s Discourse on Mathematics, 325, 327 ; Lord Bacon, 328 ; Dr. Barrow, 328, 340. Railways, Problem presented in, 361. Rainbow, Illustration, 322. RATIO -The quotient arising from dividing one number or quan tity by another, 163, 267. " Discussion concerning it, 165-171. " Arithmetical and Geometrical, 163. " How determined, 165. " An abstract number, 267, 272. " Terms direct, inverse, or reciprocal, not applicable to, 269. Reading in Addition, 116, 117 ; advantages of, 118. in Subtraction, 120. " in Multiplication, 122. in Division, 123. " of figures, its aid in practical operations, 230. Reason, To make use of arguments, 42. " A premiss placed after the conclusion, 48. REASONING.. The act of proceeding from certain judgments to another, founded on them, 9. " Three operations of the mind concerned in, 6. " Process, sameness of the, 42, 43, 45, 314. 24 370 INDEX. Reasoning processes of mathematics consist of two parts, Section 73. " in Analysis is based on the supposition that we are deal ing with things, 2*78. Reciprocal or Inverse Proportion, 269. RECTANGLE A parallelogram whose angles are right angles, 242. Remarks, Concluding subject of Arithmetic, 236. Reservoirs, Croton, description of, 362. Right angle Definition of, 258. Roman Table, when taught, 215. Root, Symbol for the extraction of, 295. Rule of Three, Solution of questions in, 169. Comparison of numbers, 186. " should precede its applications, 187. Rules, Every thing done according to, 21. of reasoning analogous to those of Arithmetic, 45 " Advantages of logical, 50. " for teaching, 186. How framed, 297. Scale of Tens, Units increasing by, 124-130, 157, 183. SCIENCE In its popular sense means knowledge reduced to order, 21, 326. " In its technical sense means an analysis of the laws of nature, 21. " contrasted with art, 22. " of Arithmetic, 172. " Principles of, 200, 208. " Methods of, must be followed in Arithmetic, 228. of Geometry, 237, 248, 257. " Objects and means of pure, 322. " should be made as much deductive as possible, 336. " Deductive and experimental, 337. " when experimental, 338, 339 ; when deductive, 338, 339. " What it has accomplished, 348. " Practical value of, in factories, 358. " " " " in constructing steamships, 359. " " " " in laying out and measuring land, 360. " " in constructing railways, 361. " Its power illustrated in Croton aqueduct, 362. " What constitutes it, 372. INDEX. 371 Second Arithmetic, its place and construction, Section 227-230. Sextant, its uses in Navigation, 359. SHADES, SHADOWS, AND PERSPECTIVE An application of Descriptive Geom etry, 381. SIGNIFICATE An individual for which a common term stands, 15. Signs, Five used to denote operations on quantity, 289. " How to be interpreted, 290. " do not affect the nature of the quantity, 290, 296. " indicate operations, 296, 298. SOLID A portion of space having three dimensions, 85. " A portion of space combining the three dimensions of length, breadth, and thickness, 246, 320 " Limit of, 247. " First idea of, how impressed, 320. Solids bounded by plane and curved surfaces, 85. " Three classes of, 246. " Analysis of comparison, 271, 272. " Comparison of, under the supposition of changes in their volumes, 272. Solution of all questions in the Rule of Three, 169. of an equation in Algebra, 308. SPACE Is indefinite extension, 81, 82. " has three dimensions, length, breadth, and thickness, 82 " Clear conception of, necessary to understand Geometry, 238. SPECIES One of the divisions of a genus in which the characteris tic is less extensive, but more full and complete, 16, 17. SUBSPECIES One of the divisions of a species, in which the characteristic is less extensive, but more full and complete, 16, 19. LOWEST SPECIES A species which cannot be regarded as a genus, 17. Spelling, 113; in Addition, &c., 115-123. SQUARE A quadrilateral whose sides are equal, and angles right angles, 242. Statement of a proposition in Algebra, 308. " in what it consists, 309. Steamship, an application of science, 359. SUBJECT The name denoting the person or thing of which some thing is affirmed or denied, 38. 372 INDEX. Subjects, How presented in a text-book, Section 209-212. Subtraction, Readings in, 120. Examples in, 152. Combinations in, 194. All operations in, governed by one principle, 232 " in Algebra, illustration of, 298. Suggestions for teaching Geometry, 273. for teaching Algebra, 315. Sum, Its definition, 203. SURFACE A portion of space having two dimensions, 84, 240, 319. " Plane and Curved, 84, 240. Surfaces, Curved, 245. " of Elementary Geometry, 245. " Limits of, 247. SURVEYING The application of the principles of Trigonometry to the measurement of portions of the earth s surface, 379. A branch of practical science, 360. SYLLOGISM A form of stating the connection which may exist for the purpose of reasoning, between three proposi tions, 40. " A formula for ascertaining what may be predicated. How it accomplishes this, 41. not meant by Aristotle to be the form in which arguments should always be stated, 53. not a distinct kind of argument, 54. " an argument stated at full length, 56. " Symbols used for the terms of, 56. Rules for examining syllogisms, 67. " has three and only three terms, 67. " " " " " propositions, 67. " test of deductive reasoning, 72, 99, 307. SYMBOLS The letters which denote quantities, and the signs which indicate operations, 87, 93, 296. used for the terms of a syllogism, 56. Advantages of, 57. Validity of the argument still evident, 58. Truths inferred by means of, true of all things, 277. regarded as things 278. Two classes of, in analysis, 29& Abstract and concrete quantity represented by, 321. INDEX. 373 SYNTHESIS The process of first considering the elements separately, then combining them, and ascertaining the results of combination, Sections 89, 327. Synthetical form, for what best adapted, 71, 89. Tables of Denominate Numbers, fractions occur five times in, 190. TECHNICAL Particular and limited sense, 91. TERM Is an act of apprehension expressed in words, 15. A singular term denotes but a single individual, 15. " A common denotes any individual of a whole class, 15. " affords the means of classification, 16. Nature of, 20. " No real thing corresponding to, 20. " Why applicable to several individuals, 20. MAJOR TERM The predicate of the conclusion, 40. MINOR TERM The subject of the conclusion, 40. MIDDLE TERM The common term of the two premises, 40. DISTRIBUTED A term is distributed when it stands for all its significates, 61. " NOT DISTRIBUTED When it stands for a part of its sig nificates only, 61. TERMS Two of the three parts of a proposition, 38. " The antecedent and consequent of a proportion, 164, 267. " should always be used in the same sense, 170, 205. TEXT-BOOK Should be an aid to the teacher in imparting instruction, and to the learner in acquiring knowledge, 209. THICKNESS A dimension of space, 82. Third Arithmetic, Principles contained in, and method of construction, 231-236. Time, Measure of, its units and scale, 148. Topography, Its uses, 360. TRAPEZOID A quadrilateral, having two sides parallel, 242. TRIANGLE A portion of a plane bounded by three straight lines, 241. " The simplest plane figure, 241. Different kinds of, 241. * regarded as a genus, 256. TRIGONOMETRY ....An application of the principles of Arithmetic, Algebra, and Geometry to the determination of the sides and angles of triangles, 378. " Plane and Spherical, 378. 374 INDEX. Troy Weight, Its units and scale, Section 137. TRUTH An exact accordance with what has been, is, or shall be, 24. Two methods of ascertaining, 24. is inference from facts or other truths, 24, 25. " regarded as a species, 25. How inferred from facts, 26. " A true proposition, 36. TRUTHS INTUITIVE OR SELF-EVIDENT Are such as become known by considering all the facts on which they depend, an 1 apprehending the relations of those facts at the s::me time, and by the same act by which we apprehend the facts themselves, 27. " LOGICAL Those inferred from numerous and complicated facts ; and also, truths inferred from truths, 28. " of Geometry, 237. " Three classes of, 237. " Demonstrative, 237. Unit fixed by the place of the figure, 127. " of the fraction, 160, 161. " of the expression, 160. Unities Advantages of the system of, 150-154. UNIT OF MEASURE . .The standard for measurement, 94. " for lines, surfaces, solids, 249. " only basis for estimating quantity, 251. UNIT ONE A single thing, 104. " All numbers come from, 108, 109, 132, 150. Method of impressing its values, 133. " Three kinds of operations performed upon, 182-186. Units, Abstract or simple, 111, 132. " Denominate or Concrete, 111. " of currency, 132. " of weight, 132. " of measure, 132, 139, 249. of length, 140. " of surface, 141. Duodecimal, 142. " of solidity, 145. " Fractional, 155, 185. INDEX. 375 UNTTY UNIT Any thing regarded as a whole, Sections 109, 110 Universal Proposition, 62. Utility and Progress, leading ideas, page 11. VARIABLES Quantities which undergo certain changes of value, the laws of which are indicated by the algebraic expres sions into which they enter, 282, 283, 313. " represented by the final letters of the alphabet, 284. Variations, Theory of, 285. Varying Scales, Units increasing by, 131, 183. Velocity known by measurement, 95. Weight known by measurement, 95. " A, should be exhibited to give ideas of numbers, 133. " Standard for, 254. Whateley, Archbishop, his views of logic, 72. Words, Definition of, 113. " expressing results of combinations, 193-197. " Double or incomplete sense of, 322. ZERO The limit of a decreasing quantity, 302-306- THE END. A. S. BARNES t COMPANY S PUBLICATIONS. Davits System of Mathematics. MATHEMATICAL WORKS, IN A SERIES OF THREE PARTS! ARITHMETICAL, ACADEMICAL, AND COLLEGIATE. BY CHARLES DAYIES, Lt.I) I. THE ARITHMETICAL COURSE FOR SCHOOLS. 1. PRIMARY TABLE-BOOK. 2. FIRST LESSONS IN ARITHMETIC. 3. SCHOOL ARITHMETIC. (Key separate.) 4. GRAMMAR OF ARITHMETIC. II. THE ACADEMIC COURSE. 1. THE UNIVERSITY ARITHMETIC. (Key separate.) 2. PRACTICAL GEOMETRY AND MENSURATION. 3. ELEMENTARY ALGEBRA. (Key separate.) 4. ELEMENTARY GEOMETRY. 5. ELEMENTS OF SURVEYING. III. THE COLLEGIATE COURSE. 1. DAVIES BOURDON S ALGEBRA. 2. DAVIES LEGENDRE S GEOMETRY AND TRIGONOMETRY. 3. DAVIES ANALYTICAL GEOMETRY. 4. DAVIES DESCRIPTIVE GEOMETRY. 5. DAVIES SHADES, SHADOWS, AND PERSPECTIVE. 6. DAVIES DIFFERENTIAL AND INTEGRAL CALCULUS. . DAVIES LOGIC AND UTILITY OF MATHEMATICS. This series, combining all that is most valuable in the various methods of European instruction, impcoved and maturd by the suggestions of more than thirty years expe rience, now forms the only complete consecutive course of Mathematics. It* methods, narmsnizir.g as the works of one mind, carry the student onward by the same analogies uid the same laws of association, and are calculated to impart a comprehensive knowl edge of the science, combining clearness in the several branches, and unity and propor tion in the whole. Being the system so long in use at West Point, through which so many men, eminent for their scientific attainments, have passed, and hf ing been adopted, as Text Books, by most of the colleges in the United States, it may be justly regarded as our NATIONAL SYSTEM OF MATHEMATICS. A. S. BARNES & COMPANY S PUBLICATIONS. Chambers Educational Court* . CHAMBERS EDUCATIONAL COURSE. THE SCIENTIFIC SECTION, The Messrs. Chambers have employed the first professors in Scotland in the prepara tion of these works. They are now offered to the schools of the United States, under the American revision of D. M. REESE, M.D., LL.D., late Superintendent of Public Schools in the city and county of New York. I. CHAMBERS TREASURY OF KNOWLEDGE, II. CLARK S ELEMENTS OF DRAWING AND PERSPECTIVE. III. CHAMBERS ELEMENTS OF NATURAL PHILOSOPHY. IV. REID & BAIN S CHEMISTRY AND ELECTRICITY. V. HAMILTON S VEGETABLE AND ANIMAL PHYSIOLOGY VI. CHAMBERS ELEMENTS OF ZOOLOGY. VII. PAGE S ELEMENTS OF GEOLOGY. "It is well known that the original publishers of these works (the Messrs. Chambers of Edinburgh) are able to command the best talent in the preparation of their books, and that it is their practice to deal faithfully with the public. This series will not dis appoint the reasonable expectations thus excited. They are elementary works pre pared by authors in every way capable of doing justice to their respective undertakings, and who have evidently bestowed upon them the necessary time and labor to adapt them to their purpose. We recommend them to teachers and parents with confidence. If not introduced as class-books in the school, they may be used to excellent advantage in gemral exercises, and occasional class exercises, for which every teacher ought to provide himself with an ample store of materials. The volumes may be had separate ly ; and the one first named, in the hands of a teacher of the younger classes, might furnish an inexhaustible fund of amusement and instruction. 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Ctn Gazette. u We notice these works, not merely because they are school books, but for the pur pose of expressing our thanks, as the advocate of the educational interenU of the people and their children, to the enterprising publishers of these and many other val uable works of the same character, the tendency of which is to diffuse useful know ledge throughout the masses, for the good work they are doing, and the hope thai thoir reward may be commensurate with their deserts," Maine School jtdvvnati. A. S. BARNES <k COMPANY S PUBLICATIONS. Parker s Natural Philosophy. NATURAL AND EXPERIMENTAL PHILOSOPHY, FOR SCHOOLS AND ACADEMIES. BY R. G. PARKER, A. M., .*/or of u Rhetorical Reader," "Exercises in English Composition," "Oullinet of History," etc., etc. I. PARKER S JUVENILE PHILOSOPHY. II. PARKER S FIRST LESSONS IN NATURAL PHILOSOPHY. III. PARKER S SCHOOL COMPENDIUM OF PHILOSOPHY. The use of school apparatus for illustrating and exemplifying the principles of Natural and Experimental Philosophy, has, within the last few years, become so general as to render necessary a work which should combine, in the same course of instruction, the theory, with a full description of the apparatus necessary for illustration and experiment. The work of Professor Parker, it is confidently believed, fully meets that requirement. It is also very full in the general facts which it presents clear and concise in its style and entirely scientific and natural in its arrangement. " This work is better adapted to the present state of natural science than any other similar production with which we are acquainted." Wayne Co. Whig. " This is a school-book of no mean pretensions and no ordinary value." Albany Spectator. " We predict for this valuable and beautifully-printed work the utmost success." Newark Daily Advertiser. " The present volume strikes us as having very marked merit." JV. Y. Courier. " It seems to me to have hit a happy medium between the too simple and the two abstract." B. A. Smith, Principal of Leicester Academy, Mass. " I have no hesitation in saying that Parker s Natural Philosophy is the most valuable elementary work I have seen." Gilbert Langdon Hmne, Prof. JVat. Phil. JV. Y. City. * I am happy to say that Parker s Philosophy will be introduced and adopted in Victoria College, at the commencement of the next collegiate year in autumn ; and I hope that will be but the commencement of the use of so valuable an elementary work in our schools in this country. The small work of Parker s (Parker s First Lessons) was introduced the last term in a primary class of the institution referred to, and that with great success. I intend to recommend its use shortly into the model school in this city, uud the larger work to the students of the provincial Normal School." E. Ryerson, Superintendent of Public Instruction of Upper Canada. " I have examined Parker s First Lessons and Compendium of Natural and Expert- mental Philosophy, and am much pleased with them. I have long felt dissatisfaction with the Text-Books on this subject most in use in this section, and am happy now to find books that I can recommend. I shall introduce them immediately into my school." Hiram Orcutt, Principal of Tlietford Academy, Vermont. " I have no hesitation in pronouncing it the best work on the subject now published, We shall use it here, and I have already secured its adoption in some of the high- schools and academies in our vicinity." M. D. Leggett, Sup. of Warren Public Schools. " We are glad to see this little work on natural philosophy, because the amount of valuable information under all these heads, to be gained from it by any little boy or girl, is Inestimable. It puts them, too, upon the right track after knowledge, and pre vents tbeir minds from being weakened and wasted by the sickly sentimentality of tales, novels, and poetry, which will always occupy the attention of the mind whn w/hing more useful has taken possession of it." Mississippian. A. S. BARNES & COMPANY S PUBLIC .TIONS. Willard s School Histories and Charts. MRS. EMMA WILLARD S SERIES OF SCHOOL HISTORIES AND CHARTS. I. WILLARD S UNIVERSAL HISTORY IN PERSPECTIVE. $1.50. II. WILLARD S TEMPLE OF TIME. Mounted, $1.25. Bound, 75 eta III. WILLARD S HISTORIC GUIDE. SOcts. IV. WILLARD S ENGLISH CHRONOGRAPHER, WILLARD S UNITED STATES. The Hon. Dan. Webster says, of an early edition of the above work, in a letter to the author, "I KKKP IT NEAR ME, AS A BOOK OF REFERENCE, ACCURATE IN FACTS AND DATES " "THE COMMITTEE ON BOOKS OF THE WARD SCHOOL ASSOCIATION RESPECTFULLY REPORT : "That they have examined Mrs. Willard s History of the United States with peculiar interest, and are free to say, that it is in their opinion decidedly the best treatise on this interesting subject that they have seen. As a school-book, its proper place is among the first. The language is remarkable for simplicity, perspicuity, and neatness ; youth could not be trained to a better taste for language than this is calculated to im part. It places at once, in the hands of American youth, the history of their country from the day of its discovery to the present time, and exhibits a clear arrangement of all the great and good deeds of their ancestors, of which they now enjoy the benefits, and inherit the renown. The struggles, sufferings, firmness, and piety of the first settlers are delineated with a masterly hand." Extract from a Report of the Ward School Teachers Association of the City of New York. " We consider the work remarkable one, in that tt forms the best book for general reading and reference published, and at the same time has no equal, in our opinion, as a text-book. On this latter point, the profession which its author has so long followed with such signal success, rendered her peculiarly a fitting person to prepare a text book." Boston Traveller. u MRS. WILLARD S SCHOOL HISTORY OF THE UNITED STATES. It is one of those rare things, a good school-book ; infinitely better than any of the United Slaes Historww fitted for schools, which we have at present." Cincinnati Gazette. * We think we are warranted in saying, that it is better adapted to meet the wants of our schools and academies in which history is pursued, than any other work of the kind now before the public. The style is perspicuous and flowing, and the prominent points of our history are presented in such a manner as to make a deep and lasting impression on the mind. We could conscientiously say much more in praise of this book, but must content ourselves by heartily commending it to the attention of those who are anxious to find a gc od text-book of American history for the use of schools." tfevburyport Watchman. I. WILLARD S HISTORY OF THE UNITED STATES, OR RE PUBLIC OF AMERICA. 8vo. Price $1.50. II. WILLARD S SCHOOL HISTORY OF THE UNITED STATES. 63 cts. III. WILLARD S AMERICAN CHRONOGRAPHER. $1.50. A. S. BARNES Sc. COMPANY S PUBLICATIONS. Parker s Rhetorical Reader. PARKER S RHETORICAL READER. 12mo. Exercises in Rhetorical Reading, designed to familiarize readers with the pauses and other marks in general use, and lead them to the practice of modulation and inflection of the voice. By R. G. PARKER, author of " Ex ercises in English Composition," " Compendium of Natural Philosophy," &c., &c. This work possesses many advantages which commend it to favor, among which ar* the following: It is adapted to all classes and schools, from the highest to the lowest. It contains a practical illustration of all the marks employed in written language : also lessons for the cultivation, improvement, and strengthening of the voice, and instructions as well as exercises in a great variety of the principles of Rhetorical Reading, which cannot fail to render it a valuable auxiliary in the hands of any teacher. Many of the exercises are of sufficient length to afford an opportunity for each member of any class, however numerous, to participate in the same exercise a feature which renders it convenient to examining committees. The selections for exercises in reading are from the most approved sources, possessing a salutary moral and religious tone, without the slightest tincture of sectarianism. " I have to acknowledge the reception through your kindness of several volumes. 1 have not as yet found time to examine minutely all the books. Of Mr. Parker s Rhe torical Reader, however, I am prepared to speak in the highest terms. I think it so well adapted to the wants of pupils, that I shall introduce it immediately in the Acad emy of which I am about to take charge at Madison, in this state. It is the best thing of the kind I have yet found. I cannot say too much in its favor." John O. Clark* Rector of the Madison Male Academy, Athens, Oa. " Mr. Parker has made the public his debtor by some of his former publications especially the Aids to English Composition and by this he has greatly increased the obligation. There are reading books almost without number, but very lew of them pretend to give instructions how to read, and, unluckily, few of our teachers are competent to supply the defect. If young persons are to be taught to read well, il must generally be done in the primary schools, as the collegiate term affords too little time to begin and accomplish that work. We have seen no other Reader with which we have been so well pleased ; and as an evidence of our appreciation of its worth, we shall lay it aside for the use of a certain juvenile specitieu of humanity in wh0se affairs we are specially interested." Christian Advocate. " We cannot too often urge upon teachers the importance of rsading, as a part of education, and we regard it as among the auspicious signs of the times, that so much more attention is given, by the best of teachers, to the cultivation of a power which ia at once a most delightful accomplishment, and of the first importance as a means of discipline and progress. In this work, Mr. Parker s volume, we are sure, will be found a valuable aid." Vermont Chronicle. "The title of this work explains its character and design, which are well carried out by the manner in which it is executed. As a class-book for students in elocution, or as an ordinary reading book, we do not think we have seen any thing superior. The dis tinguishing characteristic of its plan is to assume some simple and i.amiliar example, which will be readily understood by the pupil, and which Nature will tell him how to deliver properly, and refer more difficult passages to this, as a model. There is, how ever, another excellence in the work, which we take pleasure in commending; it is the progressiveness with which the introductory lessons are arranged. In teaching every art and science this is indispensable, and in none more so than in that of elocu tion. The pieces for exercise in reading are selected with much taste and judgment We have no doubt that those who MSB this book will be satisfied with its nicceM." Teach-*** Advocate. q A. S. BARNES & COMPANY S PUBLICATIONS. Brooks s Greek and Latin Classics. PROFESSOR BROOKS S GEEEK AND LATIN CLASSICS. THIS series of the GREEK and LATIN CLASSICS, by N. C. Brooks, of Baltimore, is on an improved plan, with peculiar adaptation to the wants of the American student. To secure accuracy of text in the works that are to appear, the latest and most approved European editions of the different classical authors will be consulted. Original illus trative and explanatory notes, prepared by the Editor, will accompany the text. These notes, though copious, will be intended to direct and assist the student in his labors, rather than by rendering every thing too simple, to supersede the necessity of due exertion on his own part, and thus induce indolent habits of study and reflection, and feebleness of intellect. In the notes that accompany the text, care will be taken, on all proper occasions, to develop and promote in the mind of the student, sound principles of Criticism, Rhetoric, History, Political Science, Morals, and general Religion; so that he may con template the subject of the author he is reading, not within the circumscribed limits of a mere rendering of the text, but consider it in all its extended connections and thus learn to think, as well as to translate. BROOKS S FIRST LATIN LESSONS. This is adapted to any Grammar of the language. It consists of a Grammar, Reader, tud Dictionary combined, and will enable any one to acquire a knowledge of the ele ments of the Latin Language, without an instructor. It has already passed through five editions. 18mo. BROOKS S C/ESAR S COMMENTARIES. (In press.) This edition of the Commentaries of Caesar on the Gallic War, besides critical and explanatory notes embodying much information, of an historical, topographical, and military character, is illustrated by maps, portraits, views, plans of battles, &c. It has a good Clavis, containing all the words. Nearly ready. 12mo. BROOKS S OVID S METAMORPHOSES. 8vo. This edition of Ovid is expurgated, and treed from objectionable matter. It is eluci dated by an analysis and explanation of the fables, together with original English notes, historical, mythological, and critical, and illustrated by pictorial embellishments ; with a Clavis giving the meaning of all the words with critical exactness. Each fable con tains a plate from an original design, and an illuminated initial letter. BROOKS S ECLOGUES AND GEORGICS OF VIRGIL. (In press.) This edition of Virgil is elucidated by copious original notes, and extracts from ancient and modern pastoral poetry. It is illustrated by plates from original designs, and contains a Clavis giving the meaning of all the words. 8vo. BROOKS S FIRST GREEK LESSONS. 12mo. This Greek elementiiry is on the same plan as the Latin Lessons, and affords equal facilities to the student. The paradigm of the Greek verb has been greatly simplified anl valuable exercises in comparative philology introduced. BROOKS S GREEK COLLECTANEA EVANGELICA. 12mo. This consists of portions of the Four Gospels in Greek, arranged in Chronological order ; and forms a connected history of the principal events in the Saviour s life and ministry. It contains a Lexicon, and is illustrated and explained by notes. BROOKS S GREEK PASTORAL POETS. (In press.) This contains the Greek Idyls of Theocritus, Bion, and Moschus, elucidated by note* and copious extracts from ancient and modern pastoral poetry. Each Idyl is illustrated by beautiful plates from original designs. It contains a good Lexicon. \. S. BARNES & COMPANY S PUBLICATIONS. Page s Theory and Practice of Teaching. THEORY AND PRACTICE OF TEACHING \ OR THE MOTIVES OF GOOD SCHOOL-KEEPING. BY DAVID PAGE, A.M., tATI PRINCIPAL OF THE STATE NORMAL SCHOOL, NEW YORK. " I received a few days since your Theory and Practice, &c., and a capital theory End capital practice it is. I have read it with immingled delight. Even if I should look through a critic s microscope, I should hardly find a single sentiment to dissent from, and certainly not one to condemn. The chapters on Prizes and on Corporal Punishment are truly admirable. They will exert a most salutary influence. So of the views sparsim on moral and religious instruction, which you so earnestly and feelingly insist upon, and yet within true Protestant limits. IT is -A GRAND BOOK, AND I THANK HEAVEN THAT YOU HAVE WRITTEN IT." Hon. Horace Mann, Secretary of the Board of Education in Massachusetts. u Were it our business to examine teachers, we would never dismiss a candidate without naming this book. Other things being equal, we would greatly prefer a teacher who has read it and speaks of it with enthusiasm. In one indifferent to such a work, we should certainly have little confidence, however he might appear in other respects. Would that every teacher employed in Vermont this winter had the spirit of this book in his bosom, its lessons impressed upon his heart!" Vermont Chronicle. "I am pleased with and commend this work to the attention of school teachers, and those who intend to embrace that most estimable profession, for light and instruction to guide and govern them in the discharge of their delicate and important duties." JV. S. Benton, Superintendent of Common Schools, State of New York. Hon. S. Young- says, "It ia altogether the best book on this subject 1 have erer seen." President North, of Hamilton College, says, " I have read it with all that absorbing self-denying interest, which in my younger days was reserved for fiction and poetry. I am delighted with the book." Hon. Marcus S. Reynolds says, " It will do great good by showing the Teacher what should be his qualifications, and what may justly be required and expected of him." "I wish you would send an agent through the several towns of this State with Page s Theory and Practice of Teaching, or take some other way of bringing this valuable book to the notice of every family and of every teacher. I should be rejoiced to see the principles which it presents as to the motives and methods of good school- keeping carried ut in every school-room ; and as nearly as possible, in the style in which Mr. Page illustrates them in his own practice, as the devoted and accomplished Principal of your State Normal School." Heni~y Barnard, Superintendent of Common Schools for the State of Rhode Island. "The Theory and Practice of Teaching, by D. P. Page, is one of the best books of the kind 1 have ever met with. In it the theory and practice of the teacher s duties are clearly explained and happily combined. The style is easy and familiar, and the suggestions it contains are plai-n, practical, and to the point. To teachers especially it will furnish very important aid in discharging the duties of their high and responsible profession." Roger S. Hoioard, Superintendent of Common Schools, Orangt Co., Vi. A. S. BARNES A; COMPANY S PUBLICATIONS. Science of the English Language CLARK S NEW ENGLISH GRAMMAR. Practical Grammar, in which WORDS, PHRASES, and SENTENCES are classi fied, according to their offices, and their relation to each other: illustrated by a complete system of Diagrams. By S. W. CLARK, A. M. Price 50 cts. "It is a most capital work, and well calculated, if we mistake not, to supersede, even in our best schools, works of much loftier pretension. The peculiarity of ita method grew out of the best practice of its author (as he himself assures us in its preface) while engaged in communicating the science to an adult class ; and his success was fully commensurate with the happy and philosophic design he has unfolded." Rahway Register. "This new work strikes us very favorably. Its deviations from older books of the kind are generally judicious and often important. We wish teachers would examine it." JVcto York Tribune. " It is prepared upon a new plan, to meet difficulties which the author has encoun tered in practical instruction. Grammar and the structure of language are taught throughout by analysis, and in a way which renders their acquisition easy and satisfac tory. From the slight examination, which is all we have been able to give it, we are convinced it has points of very decided superiority over any of the elementary works in common use. We commend it to the attention of all who are engaged in instruc tion." New York Courier and Enquirer. " From a thorough examination of your method of teaching the English language, I am prepared to give it my unqualified approbation. It is a plan original and beau tiful well adapted to the capacities of learners of every age and stage of advance ment." A. R. Simmons, Ex-Superintendent of Bristol. "I have, under my immediate instruction in English Grammar, a class of more than fifty ladies and gentlemen from the Teachers Department, who, having studied the grammars in common use, concur with me in expressing a decided preference for Clark s New Grammar, which we have used as a text-book since its publication, and which will be retained as such in this school hereafter." Professor Brittan, Principal of Lyons Union School. "Clark s Grammar I have never seen equalled for practicability, which is of the ut most importance in all school-books." 5. B. Clark, Principal of Scarborough Jlcadr emy, Maine. "The Grammar is just such a book as I wanted, and I shall make it the text-book in my school." William Brickley, Teacher at Canastota, JV*. Y. "This original production will, doubtless, become an indispensable auxiliary to re store the English language to its appropriate rank in our systems of education. After a cursory perusal of its contents, we are tempted to assert that it foretells the dawn of a brighter age to our mother tongue." Southern, Literary Gazette. " I have examined your work on Grammar, and do not hesitate to pronounce it su perior to any work with which I am acquainted. I shall introduce it into the Mount Morris Union School at the first proper opportunity." //. O. Winslow, Jl. J\L, Princi pal of Mnunt Morris Union School. "Professor Clark s new work on Grammar, containing Diagrams illustrative of his system, is, in my opinion, a most excellent treatise on the Science of the English Lan guage. The author has studiously and properly excluded from hi s book the technical ities, jargon, and ambiguity which so often render attempts to teach grammar unpleas ant, if not impracticable. The inductive plan which he has adopted, and of which he Is, in teaching grammar, the originator, is admirably adapted to t,le great purposes of both teaching and learning the important science of our language." S. JV. Sweet, Au thor of "Sweet s Elocution. 1 P LOAN DEPT