{) J/ / / 7 2 /t &M Maf/iejwaftcal Seieflce liaiies I)avies,LL.D. _ OP THE Oft %nnm\% *\ <$aKf 58 Analysis 59 Different Branches of Mathematics 60 Signs of Operation 61 Remarks on Language 62-66 SECTION III. SCIENCE OF NUMBERS. First Notions of Numbers 67 Three Forms of Language 68, 69 CONTENTS. 11 SECTION Ideas of Numbers Generalized 70, 71 Unity and a Unit Defined 72 Alphabet Defined , 73 Arithmetical Alphabet , 74 Spelling and Reading in Addition 75 General Reading 76-79 Spelling and Reading in Subtraction 80, 81 Spelling and Reading in Multiplication 82 Spelling and Reading in Division ". 83 Units increasing by Scale of Tens 84-88 Scales Uniform and Varying 89 Units increasing by Scale of Tens 90, 91 Units increasing by Varying Scales 92, 93 Integral Units Different Kinds 94, 95 Abstract Units Defined 96, 97 Units of Currency ; ; 98-100 Units of Weight 1 01-1 04 Units of Time 105 Units of Length 106 Units of Surface 107-110 Units of Volume 111-114 Units of Angular Measure 115 Advantages of System of Unities 116 Application to the Four Ground-Rules 117-119 Logical Form for Addition 120 Logical Form for Subtraction 121 Forms for Multiplication and Division 122-123 METRIC SYSTEM. Naming Measurement of Length 124, 125 Square Measure 126 Measures of Volumes 127 Dry Measure 128 12 CONTENTS. BECTIOK Liquid Measure 129 Weights 130 Nature of Metric System 131 FRACTIONAL UNITS. Scale of Tens 132-134 Fractional Units in General 135-137 Advantages of Fractional Units 138, 139 RATIO AND PROPORTION. Ratio Defined, and Properties 140-144 SECTION IT. GEOMETRY. Geometry Defined, Demonstration 145 Geometrical Magnitudes 146 Lines Straight and Curved 147 Surfaces Plain and Curved 148 Plane Figures Different Kinds 149-153 Volumes Different Kinds 154 Angles 155 General Remarks 156-157 Comparison of Figures 158-164 Properties of Figures 165 Marks of what may be Proved 166 Demonstration Its Nature. . . 167-170 Two Kinds of Demonstration 171-175 Proportion of Figures 176 CONTENTS. 13 SECTION Proportion Defined 177 Inverse Proportion 178 Comparison of Figures 179 Comparison of Volumes 180 Ratio of Two Magnitudes 181 Recapitulation for Geometry 182 Suggestions to those who Teach Geometry 182 SECTION V. ANALYSIS. Analysis Defined 183 Its General Principles 183-187 Principal branches of Analysis 188 Algebra 189 Analytical Geometry 190 Analytical Trigonometry 191 Differential and Integral Calculus 192-194 SECTION VI. ALGEBRA. General Properties 195-196 How Quantities may be Increased or Diminished 197 Signs Denoting the Operations of Change 198 Symbols their Nature 199 Coefficient Its Use 200-201 Exponents Its Use 202-203 14 CONTENTS. SECTION Division Its Signs 204 Extraction of Roots Signs 205 Minus Sign 206-207 Subtraction 208 Multiplication 209 Rules for the Signs 209-211 Zero and Infinity 212-216 Nature of the Equation 217-221 Axioms and Operations 221 General Remarks ; . . . 223 Suggestions to those who Teach Algebra 225 SECTION I. LOGIC OF MATHEMATICS. DEFINITIONS OPERATIONS OP THE MIND TERMS LOGIC INDUCTIVE DEDUCTrVE SYLLOGISM. DEFINITIONS. 1. Definition is a metaphorical word, which lit- erally signifies "laying down a boundary." All de- finitions are of names, and of names only; but in some definitions, it is clearly apparent, that nothing is intended except to explain the meaning of the word; while in others, besides explaining the mean- ing of the word, it is also implied that there exists, or may exist, a thing corresponding to the word. 2. Definitions which do not imply the existence of things corresponding to the words defined, are those usually found in the Dictionary of one's own language. They explain only the meaning of the word or term, by giving some equivalent expression which may happen to be better known. Definitions 16 MATHEMATICS. which imply the existence of things corresponding to the words defined, do more than this. For example: "A triangle is a rectilineal figure having three sides." This definition does two things: 1st. It explains the meaning of the word triangle; and, 2d. It implies that there exists, or may exist, a rec- tilineal figure having three sides. 3. To define a word when the definition is to im- ply the existence of a thing, is to select from all the properties of the thing those which are most simple, general, and obvious ; and the properties must be very well known to us before we can decide which are the fittest for this purpose. Hence, a thing may have many properties besides those which are named in the definition of the word which stands for it. 4. In Mathematics, and indeed in all exact sci- ences, names imply the existence of the things which they name ; and the definitions of those names express attributes of the things ; so that no correct definition whatever, of any mathematical term, can be devised, which shall not express certain attributes of the thing corresponding to the name. Every definition of this class is a tacit assumption of some proposition which is expressed by means of the definition, and which gives to such definition its importance. ITS LOGIC. 17 5. All the reasonings in mathematics, which rest ultimately on definitions, do, in fact, rest on the intu- itive inference, that things corresponding to the words defined have a conceivable existence as subjects of thought, and do or may have proximately, an actual existence.* OPERATIONS OF THE MIND CONCERNED IN REASONING. 6. There are three operations of the mind which are immediately concerned in reasoning. 1st. Simple Apprehension ; 2d. Judgment ; 3d. Rea- soning or Discourse. 7. Simple apprehension is the notion (or concep- tion) of an object in the mind, analogous to the per- ception of the senses. It is either Incomplex or Complex. Incomplex Apprehension is of one object, or of several without any relation being perceived between them, as of a triangle, a square, or a circlet 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 are the ob- * There are four rules which aid us in framing definitions. 1st. The definition must be adequate : that is, neither too ex- tended, nor too narrow for the word defined. 18 MATHEMATICS, jects of apprehension, whether complex or ineomplex, and pronouncing that they agree or disagree with each other, or that one of them belongs or does not helong to, the other: for example: that a right-angled tri- angle and an equilateral triangle belong to the class of figures called triangles ; or. that a square is not a circle. Judgment, therefore, is either Affirmative or Negative. 9. Reasoning (or discourse) is the act of proceeding from certain judgments to another foun ded upon them (or the result of them). 10. Language affords the signs by which these operations of the mind are recorded, expressed, and communicated. It is also an instrument of thought, and one of the principal helps in all mental operations ; hence, any imperfection in the instrument, or in the mode of using it, will materially affect any result at- tained through its aid. 2d. The definition must be in itself plainer than the word de- fined, else it would not explain it. 3. The definition should be expressed in a convenient number of appropriate words. 4th. When the definition implies the existence of a thing cor- responding to the word defined, the certainty of that existence must be intuitive. ITS LOGIC. 19 11. Every branch of knowledge has, to a certain extent, its own appropriate language ; and for a mind not previously versed in the meaning and right use of the various words and signs which constitute the lan- guage, to attempt the study of methods of philosophiz- ing, would be as absurd as to attempt reading before learning the alphabet. AB 8 TK ACTION. 12. The faculty of abstraction is that power of the mind which enables us, in contemplating any object (or objects), to attend exclusively to some particular cir- cumstance belonging to it, and quite withhold our at- tention from the rest. Thus, if a person, in contem- plating a rose should make the scent a distinct object of attention, and lay aside all thought of the form, color, &c, lie would draw off, or abstract that par- ticular part ; and therefore employ the faculty of abstraction. He would also employ the same faculty in considering whiteness, softness, virtue, existence, as entirely separate from particular objects. 13. The term abstraction, is also used to denote the operation of abstracting from one or more things the particular part under consideration; and likewise to designate the state of the mind when occupied by ab- 20 MATHEMATICS. stract ideas. Hence, abstraction is used in three senses : 1st. To denote a facility or power of the mind; 2d. To denote a process of the mind ; and, 3d. To denote a state of the mind. GENERALIZATION. 14. Generalization is the process of contemplating the agreement of several objects in certain points (that is, abstracting the circumstances of agreement, disregarding the differences), and giving to all and each of these objects a name applicable to them in respect to this agreement. For example; we give the name of triangle, to every rectilineal figure having three sides ; thus we abstract this property from all the others (for, the triangle has three angles, may be equi- lateral, or scalene, or right-angled), and name the en- tire class from the property so abstracted. Generaliza- tion therefore necessarily implies abstraction; though abstraction does not imply generalization. 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," &c, are called Singular Terms. ITS LOGIC. 21 On the other hand, those terms which denote any individual of a whole class (which are formed by the process of abstraction and generalization), are called Common or general Terms. For example ; quadrilat- eral is a common term, applicable to every rectilineal plane figure having four sides; Eiver, to all rivers; and Conqueror, to all conquerors. The individuals, for which a common term stands, are called its Slgni- ficates. CLASSIFICATION. * 16. Common terms afford the means of classifica- tion ; that is, of the arrangement of objects into classes, with reference to some common and distin- guishing characteristic. A collection, comprehending a number of objects, so arranged, is called a Genus or Species genus being the more extensive term, and often embracing many species. For example : animal is a genus embracing every thing which is endowed with life, the power of volun- tary motion, and sensation. It has many species, such as man, beast, bird, &c. .If we say of an animal, that it is rational, it belongs to the species man, for this is the characteristic of that species. If we say that it has wings, it belongs to the species bird, for this, in like manner, is the characteristic of the species bird. A species may likewise be divided into classes or 22 MATHEMATICS. subspecies ; thus the species man, may be divided into the classes, male and female, and these classes may be again divided until we reach the individuals. IT. Now, it will appear from the principles which govern this system of classification, that the character- istic of a genus is of a more extensive signification, but involves fewer particulars than that of a species. In like manner, the characteristics of a species is more extensive, but less full and complete, than that of a subspecies or class, and the characteristics of these less full than that of an individual. ^^^ For example ; if we take as a genus the / 1 \ Quadrilaterals of Geometry, of which the characteristic is, that they have four sides, / v then every plane rectilineal figure, having / 2 \ four sides, will fall under this class. If, then, we divide all quadrilaterals into two V ~ \ species, viz. those whose opposite sides, taken * - A 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 irregular quadrilaterals, including the trape- zoid (1 and 2) ; and in the other, the parallel- ogram, the rhombus, 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 ITS LOGIC. 23 the one, the irregular quadrilaterals, called trapeziums (1), and in the other, the trapezoids (2); and each of these classes, being made up of individuals having the same characteristics, are not susceptible of further division. If we divide the second species into two classes, ar- ranging those which have oblique angles in the one, and those which have right angles in the other, we shall have in the first, two varieties, viz. the common parallelogram (3), and the equilateral parallelogram or rhombus (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 quadrilateral; and hence, possesses the characteristic of the genus ; and each variety of both species enjoys all the charac- teristics of the species to which it belongs, together with some other distinguishing feature of a species nearer the genus ; and similarly, of all classifications. 18. In special classifications, it is often not neces- sary to begin with the most general characteristics ; and then the genus with which we begin, is in fact but a species of a more extended classification, and is called a Subaltern Genus. For example; if w r e begin with the genus Paral- lelogram, 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 24 MATHEMATICS. there will be two varieties, viz. in the first, the common parallelogram and tlie rhombus ; and in the second, the rectangle and square. 19. A genus which cannot be considered as a spe- cies, that is, which cannot be referred to a more ex- tended classification, is called the highest genus ; and a species which cannot be considered as a genus, because it contains only individuals having the same character- istics, is called the lowest species. NATURE OF COMMON TEEMS. 20. It should be steadily kept in mind, that the "common terms" employed in classification, have not, as the names of individuals have, any real existing thing in nature corresponding to them ; but that each is merely a name denoting a certain inadequate notion which our minds have formed of an individual. But as this name does not include any thing wherein that individual differs from others of the same class, it is ap- plicable equally well to all of them, as to any of them. Thus, quadrilateral denotes no real thing, distinct from each individual, but merely any rectilineal figure of four sides, viewed inadequately / that, is abstracting and omitting all that is peculiar to each individual of the class. By this means, a common term becomes ITS LOGIC. 25 applicable alike to any one of several individuals ; or, taken in the plural, to several individuals together. In regard to classification, we should also bear in mind, that we may fix, arbitrarily, on the characteristic which we choose to abstract and consider as the basis of our classification, disregarding ail the rest : so that the same individual may be referred to any of several different species, 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, it means knowledge reduced to order ; that is, knowledge so classified and arranged as to be easily remembered, readily referred to, and advantageously applied. In a more strict and technical sense, it has another significa- tion. "Every thing in nature, as well in the inanimate as in the animated world, happens or is done according to rules, though we do not always know them. Water falls according to the laws of gravitation, and the mo- tion of walking 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 can only say that, in this case, the rules are unknown to us," 2 26 MATHEMATICS Assuming tliat all the, p] ] en omen a of nature are con- sequences of general and immutable laws, we may de- fine Science to be the analysis of those laws compre- hending not only the connected processes of experiment and reasoning which make them known to man, but also those processes of reasoning which make known their individual and concurrent operation in the devel- opment of individual phenomena. ART. 22. Art is the application of knowledge to practice. Science is conversant about knowledge : Art is the use or application of knowledge, and is conversant about works. Science has knowledge for its object : Art has knowledge for its guide. A principle of science, when applied, becomes a rule of art. The developments of science increase knowledge : the applications of art add to works. Art, necessarily, presupposes knowledge: art, in any but its infant state, 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 human affairs, that to enable one thing to be do?ie, it is often requisite to know the nature and properties of many things. ITS LOGIC. 27 KNOWLEDGE 23. Kn<>wlfxge is a clear and certain conception of that which is true, and implies three things : 1st. Firm belief ; 2d. Of what is true ; and, 3d. On sufficient grounds. FACTS AND TRUTHS. 24. Our knowledge is of two kinds : of facts and truths. A fact is any thing that has been or is. That the sun rose yesterday, is a fact: that he gives light to-day, is a fact. That water is fluid and stone solid, are facts. We derive our knowledge of facts through the medium of the senses. Truth is an exact accordance with what has bekn, is, or shall be. There are two methods of ascertain- ing truth : 1st. By comparing known facts with each other; and, 2dly. By comparing known truths with each other. Hence, truths are inferences from facts or other truths made by a mental process called Reasoning. intuitive truths. 25. Intuitive Truths are those which become known, by considering all the facts on which they de- 28 MATHEMATICS. pend, and which are inferred the moment the facts are apprehended. The term Intuition is strictly applicable only to that mode of contemplation in which we look at 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-evident truths are those which are conceived in the mind immediately; that is, which are perfectly con- ceived by a single process, the moment the facts on which they depend are apprehended. They are neces- sary consequences of conceptions respecting which they are asserted. The axioms of Geometry afford the sim- plest and most unmistakable class of such truths. " A whole is equal to the sum of all its parts," is an intuitive or self-evident truth, inferred from facts previously learned. For example ; having learned from experience and through the senses what a whole is, and, from experiment, the fact that it may bo divided into parts, the mind perceives 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 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 conclusion is drawn, are pre- sented to the mind, and the conclusion is immediately made ; hence, it is an intuitive or self-evident truth. ITS LOGIC. 29 All the other axioms of Geometry are deduced from premises and by processes of inference, entirely similar. We would not call these experimental truths, for they are not alone the results of experiment or experience. Experience and experiment furnish the requisite in* formation, but the reasoning power evolves the gen- eral truth. LOGIC OK REASONING. 26. Logic takes note of and decides, upon the sufficiency of the evidence by which truths are estab- lished. Our assent to the conclusion being grounded on the truth of the premises, we never could arrive at any knowledge by reasoning, unless something were known antecedently to all reasoning. It is the prov- ince of Logic to furnish the tests by which all truths that are not intuitive may be inferred from the prem- ises. It has nothing to do with ascertaining facts, nor with any proposition which claims to be believed on its own intrinsic evidence. But, so far as our knowledge is founded on truths made such by evidence, that is, derived from facts or other truths previously known, whether those truths be particular truths, or general propositions, it is the province 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 grounded. And since by far the greatest portion of 30 MATHEMATICS. our knowledge, whether of particular or general truths, is avowedly matter of inference, nearly the whole, not only of science, but of human conduct, is amenable to the authority of Logic. Logic is divided into two kinds : Inductive and Deductive. INDUCTION. 27. That part of Logic which infers truths from facts, is called Induction. Inductive reasoning is the application of the reasoning process to a given number of facts, for the purpose of determining if what has been ascertained respecting one or more of the indi- viduals, is true of the whole class. Hence, Induction is not the mere sum of the facts, but a conclusion drawn from them. The logic of Induction consists in classing the facts and stating the inference from them. "28. Induction, therefore, is a process of inference. It proceeds from the known to the unknown ; and any operation involving no inference, any process in which the conclusion is a mere fact, and not a truth, does not fall within the meaning of the term. The conclusion must be broader than the premises. The premises are facts : the conclusion must be a truth. Induction, therefore, is a process of generalization. It is that operation of the mind by which we infer that ITS LOGIC. 31 what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, Induction is the process by which we conclude that what is true of certain individuals of a class, is true of the whole class; or that what is true at certain times, will be true, under similar circumstances, at all times. 29. Induction always presupposes, not only that the necessary observations are made with the necessary accuracy, but also, that the results of these observations are, so far as practicable, connected together by general descriptions ; enabling the mind to represent to itself as wholes, whatever phenomena are capable of being so represented. To suppose, however, that nothing more is required from the conception than that it should serve to con- nect the observations, would be to substitute hypothesis for theory, and imagination for proof. The connecting link must be some character which really exists in the facts themselves, and which would manifest itself there- in, if the condition could be realized which our organs of sense require. For example : Bakewell, a celebrated English cattle- breeder, observed, in a great number of individual beasts, a tendency to fatten readily, and in a great number of others the absence of this constitution : in every individual of the former description, he observed 32 MATHEMATICS. 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 dif- ferent make from the others. These facts were his data; from which, combining them with the general principle, that nature is steady and uniform in her proceedings, he logically drew the conclusion that beasts of the specified make have universally a peculiar tendency to fattening. The Induction consisted in the generalization ; that is, in inferring, from all the data, that certain circumstances would be found in the whole class. DEDUCTION. 30. We have seen that all processes of Keasoning, in which the premises are particular facts, and the conclusions general truths, are called Inductions. All processes of Reasoning, in which the premises are gen- eral truths and the conclusions particular truths, are called Deductions. Hence, a deduction is the process of reasoning by which a particular truth is inferred from other truths which are known or admitted. The formula for all deductions is found in the Syllogism, <,he parts, nature, and uses of which we shall now pro^ ceed to explain. ITS LOGIC. 33 PROPOSITIONS. 31. A proposition is a judgment expressed in words. Hence, a proposition is defined logically, " A sentence indicative :" affirming or deifying ; therefore, it must not be ambiguous, for that which has more than one meaning is in reality several propositions ; nor imperfect, nor ungrammaiical, for such expressions have no meaning at all. 32. Whatever can be an object of belief, or even of disbelief, must, when put into words, assume the form of a proposition. All truth and all error lie in propo- sitions. What we call a truth, is simply a true propo- sition ; and errors are false propositions. To know the import of all propositions, would be to know all ques- tions which can be raised, and all matters which are susceptible of being either believed or disbelieved. 33. The first glance at a proposition shows that it is formed by putting together two names. Thus, in the proposition " Gold is yellow," the property yellow is affirmed of the substance gold. In the proposition, " Franklin was not born in England," the fact ex- pressed by the words lorn in England is denied of the man Franklin. 2* 34 MATHEMATICS. 34. Every proposition consists of three parts : the Subject, the Predicate, and the Copula. The subject is the name denoting the person or thing of which something is affirmed or denied : the predicate is that which is affirmed or denied of the subject ; and these 'two are called the terms (or extremes), because, logic- ally, the subject is placed first, and the predicate last. The copula, in the middle, indicates the act of judg- ment, and is the sign denoting that there is an affirma- tion or denial. Thus, in the proposition, " The earth is round ;" the subject is the words " the earth," being that of which something is affirmed : the predicate, is the word round, which denotes the quality affirmed, or (as the phrase is) predicated : the word is, which serves as a connecting link between the subject and the predi- cate, to show that one of them is affirmed of the other, is called the Copula. The copula must be either is, or is not, the substantive verb being the only verb recog- nized by Logic. All other verbs are resolvable, by means of the verb " to be," and a participle or adjec- tive. For example : " The Romans conquered ;" the word " conquered" is both copula and predicate, being equivalent to " were victorious?'' Hence, we might write, " The Romans were victorious," ITS LOGIC. 35 in which were is the copula, and victorious the predi- cate. 35. A proposition being a portion of discourse, in which something is affirmed or denied of something, all propositions may be divided into affirmative and nega- tive. An affirmative proposition is that in which the predicate is affirmed of the subject ; as, " Csesar is dead." A negative proposition is that in which the predicate is denied of the subject ; as, " Csesar is not dead." The copula, in this last species of proposition, consists of the words "is not," which is the sign of negation ; " is" being the sign of affirmation. SYLLOGISM. 36. A syllogism is a form of stating the connection which may exist, for the purpose of reasoning, between three propositions. Hence, to a legitimate syllogism, it is essential that there should be three, and only three, propositions. Of these, two are admitted to be true. The first is called the major premise ; the sec- ond, the minor premise ; and the third, which is proved from these two, and is called the conclusion. For example : " All tyrants are detestable : Caesar was a tyrant ; Therefore, Ceesar was detestable." 36 MATHEMATICS. Now, if the first two propositions be admitted, the third, or conclusion, necessarily follows from them, and it is proved that Cjssar was detestable. Of the two terms of the conclusion, the Predicate (detestable) is called the major term, because it comes from the major premise ; the Subject (Caesar) the minor term, because it comes from the minor premise ; and these two terms, together with the term " tyrant," are alone used in the three propositions of the syllo- gism, each term being used twice. Hence, every syllogism has three, and only three, different terms. The term tyrant, common to the two premises, and with which both the terms of the conclusion were separately compared, before they were compared with each other, is called the middle term. Therefore, in the above syllogism, " All tyrants are detestable," is the major premise, and " Csesar was a tyrant," the minor premise, " tyrant" the middle term, and " Caesar was detestable " the conclusion. 37. The syllogism, therefore, is a mere formula for ascertaining what may, or what may not, be predicated of a subject. It accomplishes this end. by means of ITS LOGIC. 37 two propositions, viz. by comparing the given predicate of the first (a Major Premise), and the given subject of the second (a Minor Premise), respectively with one and the same third term (called the middle term), and thus under certain conditions, or laws of the syllogism eliciting the truth (conclusion), that the given predi- cate must be predicated of that subject. It will be seen that the Major Premise always declares, in a gen- eral way, such a relation between the Major Term and the Middle Term ; and the Minor Premise declares, in a more particular way, such a relation between the Minor Term and the Middle Term, as that, in the Con- clusion, 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. In Mathematics, the reasoning is entirely Deductive: hence, every process, in proceeding from known to un- known truths, may be reduced to the form of the Syl- logism. . o b g ^ o 3 03 3 i n 3 3 a M w W CO g 3 3 w Hi C5 < o IH Hi GQ > Besides the units of surface already considered, t&ere is another kind, called, DUODECIMAL UNITS. 108. The duodecimal units are generally used in board measure, though they may be used in all superficial measurements, and also in volumes. The square foot is the base of this, class of units, and the others are deduced from it, by a descending scale of twelve. 109. It is proved in Geometry, that if the number of linear units in the base of a rectangle be multiplied by the number of linear units in its height, the numerical value of the product will be equal to the number of superficial units in the figure. Knowing this fact, w T e often express it by saying, that " feet multiplied by feet give square feet," and " yards multiplied by yards give square yards." But as feet cannot be taken feet times, nor yards yard times, this language, rightly understood, is but a concise form of expression for the principle stated above. "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. INTEGRAL NUMBERS. 79 110. If now, 1 foot in length be multiplied by 1 inch = T \ of 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 pro- duct will be three-twelfths of a square foot; and similarly, for a multiplier of any number of inches. If, now, we multiply 1 inch by 1 inch, the product may be represented by 1 square inch: that is, by one- twelfth of the last unit. Hence, the units of this measure decrease according to the scale of 12. The units are, 1st. Square feet arising from multiplying feet by feet: 2d. Twelfths of square feet arising from multiply- ing the denomination of feet by the denomination of inches : 3d. Twelfths of twelfths arising from multiplying inches by inches. The same remarks apply to the smaller divisions of the foot, according to the scale of twelve. The difficulty of computing, in this measure, arises from the changes in the units. UNITS OF VOLUME. 111. Volume has been denned in Geometry to be a limited portion of space. It has already been stated, 80 MATHEMATICS. that if length be multiplied by breadth, the product may be represented by units of surface. It is also proved, in Geometry, that if the length, breadth, and height of any regular figure, of a square form, be multiplied together, the product may be represented by units of volume whose number is equal to this product. Each unit of volume is a cube constructed on the linear unit as an edge. Thus, if the linear unit be 1 foot, the unit of volume will be 1 cubic foot; that is, a cube constructed on 1 foot as an edge ; and if it be 1 yard, the unit will be 1 cubic yard. The three units, viz. the unit of length, the unit of surface, and the unit of volume, are essentially dif- ferent in kind. The first is a line of a known length ; the second, a square of a known side; and the third, a volume, called a cube, of a known base and height. These are the units used in all kinds of measure- ments of solids, excepting only the ^duodecimal system, which has already been explained. 112. A volume filled with matter, is called a solid or body / and for this, the unit of measure is the cube, except in the case of wood, where it is a cord of 128 cubic feet. But a given volume filled with a liquid, takes the name of a gallon ; and a larger volume filled with grain, takes the name of a bushel. Hence, the Liquid and Dry measures, are measures of volume, and differ INTEGRAL NUMBERS. 81 only from the measurement of solidity, in the form and value of the unit of measure. LIQUID MEASURE. 113. The units of Liquid Measure are, 1 gill, 1 pint, 1 quart, 1 gallon, 1 barrel, 1 hogshead, 1 pipe, 1 tnn. The scale is a varying scale. Its steps, in pass- ing from the unit of the lowest denomination, are, four, two, four, thirty-one and a half, sixty-three, two, and two. DRY MEASURE. 114. The units of this measure are, 1 pint, 1 quart, 1 peck, 1 bushel, and 1 chaldron. The steps of the scale, in passing from units of the lowest de- nomination, are, two, eight, four, and thirty -six. ANGULAR MEASURE. 115. The units of this measure are, 1 second, 1 minute, 1 degree, 1 sign, 1 circle. The steps of the scale, in passing from units of the lowest denomina- tion to those of the higher, are sixty, sixty, thirty, and twelve. A* 82 MATHEMATICS. ADVANTAGES OF THE SYSTEM OF UNITIES. 116. It may well be asked, if the method here adopted, of presenting the elementary principles of /arithmetic, has any advantages over those now in general use. It is supposed to possess the following : 1st. The system of unities teaches an exact analy- sis of all numbers, and unfolds to the mind the different ways in which they are formed from the unit one, as a base. 2d. Such an analysis enables the mind to form a definite and distinct idea of every number, by pointing out the relation between it and the unit from which it was derived. 3d. By presenting constantly to the mind the idea of the unit one, as the base of all numbers, the mind is insensibly led to compare this unit with all the numbers which flow from it, and then it can the more easily compare those numbers with each other. 4th. It affords a more satisfactory analysis and a better understanding of the four ground-rules, and indeed of all the operations of arithmetic, than any other method of presenting the subject. FOUR GROUND-RULES. 117. Let us take the two following examples in Addition, the one in abstract and the other in INTEGRAL NUMBERS. 83 denominate numbers, and then analyze the process of finding the sum in each. SIMPLE NUMBERS. DENOMINATE NUMBERS. 874198 cwt. qr. lb. oz. dr. 86984 3 3 24 15 14 3641 6 3 23 14 8 914823 10 3 23 14 6 In both examples we begin by adding the units of the lowest denomination, and then, we divide their sum by so many as make one of the denom- ination next higher. We then set down the re- mainder, and add the quotient to the units of that denomination. Having done this, we apply a sim- ilar process to all the other denominations the principle being precisely the same in both examples. We see, in these examples, an illustration of a general principle of addition, viz. that units of the same hind are always added together. % 118. Let us take two similar examples in Sub- traction. SIMPLE NTJMBEBS. DENOMINATE NUMBERS. 8403 ,. d . far. 3298 6 9 7 2 5105 . 3 10 8 4 2 18 10 2 84: MATHEMATICS. In both examples we begin with the units of the lowest denomination, and as the number in the subtrahend is greater than in the place directly above, we suppose so many to be added in the minuend as make one unit of the next higher de- nomination. We then make the subtraction, 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 em- ployed is the same in both examples. Also, that units of any denomination in the subtrahend are taken from those of the same denomination in the minuend. 119. Let us now take similar examples in Mul- tiplication. SIMPLE NUMBERS. DENOMINATE NUMBERS. 87464 lb I 3 3 jr. 5 9 7 6 2 15 4.37220 5 48 3 2 1 15 In these examples we see, that we multiply, in succession, each order of units in the multiplicand 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 INTEGRAL NUMBERS. 85 principle of the process is therefore the same in both examples. LOGICAL FORM FOR ADDITION. 120. Def. A number which contains as many units as all the numbers added, is called their sum. With this definition, let it be required to prove tli at 8 is the sum of 3 and 5. The Syllogistic form is : A number which contains as many units as all the numbers added, is called their sum. (Major Premise.) Eight contains as many units as there are in 3 and 5; (because 3 counted on to 5 make 8). (Minor Premise.) Therefore, 8 is the sum of 3 and 5. This proof, logically, is perfect, because it brings the result of the operation performed on 3 and 5, under the term, Sum. LOGICAL FORM FOR SUBTRACTION. 121. Def. A number which added to the less of two numbers will give the greater, is called their Difference. What is the difference between 7 and 10? The syllogistic form is : A number w T hich added to the less of two num- bers will give the greater, is called their difference; 3 added to 7 gives 10 ; 86 MATHEMATICS. Therefore, 3 is the difference between 7 and 10. Here, again, the logical form merely brings the re- sult of the operation under the definition. 122. In Multiplication, if we define the operation to be, the process of taking the multiplicand as many times as there are units in the multiplier, we prove the operation by showing that the result fulfils this con- dition. 123. So, in Division, if we define the quotient to be such a number as multiplied by the divisor will produce the dividend ; we prove the operation to be correct, when we show that the number found is such a multiplier. Every proof in the entire range of mathematical science, consists in 'bringing the thing to he proved, under a definition, an axiom, or a proposition pre- viously established. METRIC SYSTEM. 124. Every system of Weights and Measures must have an invariable unit for its base and every other unit of the entire system should be derived from it, according to a fixed law. The French Government, in order to obtain an invariable unit, measured a degree of the arc of a METRIC SYSTEM. 87 meridian on the Earth's surface; and from this com- puted the length of the meridional arc from the equator to the pole. This length they divided into ten million equal parts, and then took one of these parts for the unit of length, and called it a Meter. The length of this meter is equal to 1 yard, 3 inches and 37 hun- dredths of an inch, very nearly. Thus they obtained the length of the unit which is the base of the Metric System of Weights and Measures. The next step was to fix the law by which the other units of the system should be obtained from the base. As the scale of tens is the simplest law by which we can pass from one unit to another, that scale was adopted, and the larger units are formed by multiply- ing the base continually by 10, and the smaller, by dividing it continually by 10. NAMING. 125. The names, in the ascending scale, are formed by prefixing to the base, Meter, the words Deca (ten), Hecto (one hundred), Kilo (one thousand), Myria (ten thousand), from the Greek numerals; and in the descending scale, by prefixing Deci (tenth), centi (hundredth), Milli (thousandth), from the Latin numerals. Hence, the name of a unit indicates whether it is greater or less than the standard, and also, how many times. 88 MATHE M A TICS MEASUREMENT OF LENGTH Hence, for the measurement of length, we have Ascend tfn^ #mte. fca Descending Scale. s Eh N . CD s 0) 3 a c3 | o o a o 1 O 4-> Q a o Centimeter Millimeter < a a H A a A in which the increase and decrease, from the base, takes place according to the scale of tens. SQUARE MEASURE. 126. The unit of measure for surfaces is a square described on a line 10 meters in length, and is called an Ark. Hence, the unit of surface is equal to 100 square meters. It is also equal to 4 perches, nearly. There are but two other units used in the French system the Centare, which is one hundredth of the Are, and the Hectare, which is one hundred times the Are. MEASURES OF VOLUME. 127. The unit for the measurement of volumes is the cube constructed on the decimeter, as an edge. It is called a Leter ; and is equal to 61 cubic inches, very nearly. All the other denominations are derived from METRICS Y STEM. 89 the base, in the same manner as in the measurement of length. Thus, we have Ascending Scale i i O P J M I >3 Descending Scale. Kiloleter Hectoleter tit we say, that A is directly proportional to B, and in- versely proportional to C. The terms Direct, Inverse or Keciprocal, apply to GEOMETRY. 129 the nature of the proportion, and not to the Eatio, which is always a mere quotient and the measure of proportion. The term Direct applies to all propor- tions in which the terms increase or decrease to- gether ; and the term Inverse 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 de- finition. COMPARISON OF FIGURES. 179. In comparing geometrical magnitudes, by means of their quotient, it is not the quotient alone which we consider. The comparison implies a gen- eral relation of the magnitudes, which is measured by the Eatio. For example : we say that " Similar triangles are to each other as the squares of their homologous sides.'' What do we mean by that? Just this : That the area of a triangle Is to the area of a similar triangle, As the area of a square described on a side of the first To the area of a square described on a homolo- gous side of the second. Thus, we see that every term of such a proportion is in fact a surface, and that the area of a triangle in- creases or decreases much faster than its sides ; that is, 6* 130 MATHEMATICS. 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 be nine times as great ; or if we divide each side by two, we diminish the area four times, and so on. Again : The area of one circle Is to the area of another circle, As a square described on the diameter of the first To a square described on the diameter of the second. Hence, if"\ve double the diameter of a circle, the area of the circle whose diameter is so increased will be in- creased four times; if we multiply the diameter by three, the area will be increased nine times; and simi- larly, if we divide the diameter. 180. In comparing volumes together, the same gen- eral principles obtain. Similar volumes are to each other as the cubes described on their homologous or corresponding sides. That is, A prism Is to a similar prism, As a cube described on a side of the first Is to a cube described on a homologous side of the second. Hence, if the sides of a prism be doubled, the vol- umes, or contents, will be increased eight-fold. Again : GEOMETRY. 131 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 volume, or contents, will be increased eight-fold ; if the diameter be multiplied by three, the volume, or con- tents, will be increased twenty-seven fold ; if the diame- ter be multiplied by four, the volume, or contents, will be increased sixty -four fold ; the volumes increasing as the cubes of the numbers 1, 2, 3, 4, &c. 181. The relation or ratio of two magnitudes to each other, may be, and indeed is, expressed by an ab- stract number. This number has a fixed value so long as we do not introduce a change in the volumes ; but, if we wish to express their ratio under the supposition that their volumes may change according to fixed laws (that is, so that the figures shall continue similar), we then compare them with similar figures - described on their homologous or corresponding sides ; or, what is the same thing, take into account the corresponding changes which take place in the abstract numbers that express their volumes. 132 MATHEMATICS EE CAPITULATION. 182. 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, or those limited portions of space which are called, Geo- metrical Magnitudes. 2d. That the geometrical magnitudes embrace four species of figures : 1st. Lines straight and curved ; 2d. Surfaces plane and curved ; 3d. Volumes bounded either by plane surfaces or curved, or both ; and, 4th. Angles, arising from the positions of lines and planes, and by which they are bounded. 3d. That the science of Geometry is made up of those processes by means of which all the properties of these magnitudes are examined and developed, and that the results arrived at constitute the truths of Ge- ometry. 4th. That the truths of Geometry may be divided into three classes; 1st. Those implied in the definitions, viz., that things exist corresponding to certain words defined ; 2d. Intuitive or self-evident truths embodied in the axioms; GEOMETRY. 133 3d. Truths deduced (that is, inferred) from the definitions and axioms, called Demonstrative Truths. 5th. That the examination of the properties of the geometrical magnitudes has reference, 1st. To their comparison with a standard or unit of measure ; 2d. To the discovery of properties belonging to an individual figure, and yet common to the entire class to which such figure belongs ; 3d. To the comparison, with respect to magni- tude, of figures of the same species with each other ; viz. lines with lines, surfaces with surfaces, volumes with volumes, and angles with angles. SUGGESTIONS FOR THOSE WHO TEACH GEOMETRY. 1. Be sure that your pupils have a clear apprehen- sion of space, and of the notion that Geometry is con- versant about space only. 2. Be sure that they understand the signification of the terms, lines, surfaces, volumes, and angles, and that these names indicate certain portions of space corresponding to them. 3. See that they understand the distinction between a straight line and a curve ; between a plane surface and a curved surface ; between a volume bounded by 134 MATHEMATICS planes and a volume bounded by curved surfaces ; and also, between the different kinds of angles. 4. Be careful to have them note the characteristics of the different species of plane figures, such as tri- angles, quadrilaterals, pentagons, hexagons, &c- ; and then the characteristic of each 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 character- istic differences of the volumes. Let 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 Volumes as a genus, let them give the species and subspecies into which the volumes may be divided. 6. Having thus made them familiar with the things which are the subjects of the reasoning, explain care- fully the nature of the definitions : then of the axioms, the grounds of our belief in them, and the sources from which those self-evident truths are inferred. 7. Then explain to them, that the definitions and axioms are the basis of all geometrical reasoning : 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 black- board. This will establish a connection between the GEOMETRY. 135 eye and the hand, and give, at the same time, a clear perception of the figure and a distinct apprehension of the relation of its parts. 9. Let the pupil, in every demonstration, first enun- ciate, 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, any truth is inferred from its connection with one before known, let the truth so referred to be fully and accu- rately stated, even though the number of the proposi- tion in which it is proved be also required. This is deemed important. 11. Let the pupil be made to understand that a demonstration is but a series of logical arguments arising from comparison, and that the result of every comparison, in respect to quantity, contains the mark either of equality or inequality. 12. Let the distinction between a positive and negative demonstration be fully explained and clearly apprehended. 13. In the comparison of quantities with each other, great care should be taken to impress the fact that proportion exists only between quantities of the same kind, and that ratio is the measure of proportion. 14. Do not fail to give much importance to the 136 MATHEMATICS Jcind of quantity under consideration. Let the ques- tion be often put, What kind of quantity are you considering? Is it a line, a surface, a volume, or an angle ? And what kind of a line, surface, volume or angle ? 15. In all cases of measurement, the unit of measure should receive special attention. If lines are measured, or compared by means of a common unit, see that the pupil perceives that unit clearly, and apprehends distinctly its relations to the lines which it measures. In surfaces, take much pains to mark out on the black- board the particular square which forms the unit of measure, and write unit, or unit of measure, over it. So in the measurement of volumes, let the unit or measuring cube be exhibited, and the conception of its size clearly formed in the mind; and then impress the important fact, that, all measurement consists in merely comparing a unit of measure with the quan- tity measured f 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 Equal in all the parts, and never permit the pupil to use them as synonymous. An accurate use of words leads to nice discriminations of thought. SECTION V. ANALYSIS. 183. Analysis is a general term, embracing that entire portion of mathematical science in which the quantities considered are represented by letters of the alphabet, and the operations to be performed on them are indicated by signs. 184. We have seen that all numbers must be numbers of something ; for, there is no such thing as a number without a base: 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 may be numbers of any thing, for the unit may be whatever we choose to make it. 185. All quantity consists of parts, which can be numbered exactly or approximatively, and, in this respect, possesses all the properties of number. Prop- ositions, therefore, concerning numbers, have the re- 138 MATHEMATICS markable peculiarity, that they are propositions con- cerning all quantities whatever. That half of six is three, is equally true, whatever the word six may represent, whether six abstract units, six men, or six triangles. Analysis extends the generalization still further. A number represents, or stands for, that particular number of things of the same kind, without reference to the nature of the thing; but an analytical symbol does more, for it may stand for all numbers, or for all quantities which numbers rep- resent, or even for quantities which cannot be ex- actly expressed numerically. As soon as we conceive of a thing we may con- ceive it divided into equal parts, and may represent either or all of those parts by a or x, or may, if we please, denote the thing itself by a or a?, without any reference to its being divided into parts. 186. In Geometry, each geometrical figure stands for a class : and when we have demonstrated a prop- erty of a figure, that property is considered as proved for every figure of the class. For example : when we prove that the square described on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides, we demon- strate the fact for all right-angled triangles. But in analysis, all numbers, all lines, all surfaces, all vol- umes, and all angles, may be denoted by a single ANALYSIS. 139 symbol, a or x. 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 things. The mere written characters, a, &, c, d, x, y, 2, serve as the representatives of things in general, whether abstract or concrete, whether known or unknown, whether finite or infinite. 1ST. In the nses which we make of these symbols, and the processes of reasoning carried 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 properties of things in general, without pausing to inquire what kind of thing is im- plied. Thus, we define an equation to be a proposi- tion in which equality is predicated of one thing as compared with another. For example : a + c = x, is an equation, because x is declared to be equal to the sum of a and c. In the solution of 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 will be equal." Now, 140 MATHEMATICS. these axioms do not express qualities of language, but properties of quantity. Hence, all inferences in mathematical science, deduced through the instru- mentality of symbols, whether Arithmetical, Geo- metrical, or Analytical, must be regarded as con- cerning quantity, and not symbols. Since analytical symbols are the representatives of quantity in general, there is no necessity of keeping the idea of quantity continually alive in the mind ; and the processes of thought may, without danger, be allowed to rest on the symbols themselves, and therefore become, to that extent, merely mechanical. But, when we look back and see on what the rea- soning is based, and how the processes have been conducted, we shall find that every step was taken on the supposition that we were actually dealing with things, and not symbols ; and that, without this understanding of the language, the whole system is without signification, and fails. 188. There are four principal branches of Analysis : 1st. Algebra. 2d. Analytical Geometry. 3d. Analytical Trigonometry. 4th. Differential and Integral Calculus. ANALYSIS. 14:1 ALGEBEA. 189. Algebra is, in fact, a species of universal Arithmetic, in which letters and signs are employed to abridge and generalize all processes involving numbers. It is divided into two parts, correspond- ing to the science and art of Arithmetic : 1st. That which has for its object the investiga- tion 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 in- volving the determination of certain numbers which are unknown, from their connection with certain others which are known or given. ANALYTICAL GEOMETRY. 190. Analytical Geometry examines the proper- ties, measures, and relations of the geometrical mag- nitudes by means of the analytical symbols. This branch of mathematical science originated with the illustrious Descartes, a celebrated French mathema- tician of the seventeenth century. He observed that the positions of points, the direction of lines, and the forms of surfaces, could be expressed by means of the algebraic symbols ; and consequently. 142 MATHEMATICS. ii i that every change, either in the position or extent of a geometrical magnitude, produced a correspond- ing change in certain symbols, by which such mag- nitude 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 corresponded certain analytical expressions (called their equations), it followed, that if the pro- cesses were known by which these equations could be examined, the relation of their parts determined, and the laws according to which those parts vary, relative to one another, ascertained, then the cor- responding changes in the geometrical magnitudes, thus represented, could be immediately inferred. Hence, it follows that every geometrical question can be solved, if we can resolve the corresponding algebraic equation ; and the power over the geomet- rical magnitudes was extended just in proportion as the properties of quantity were brought to light by means of the Calculus. The applications of this Cal- culus were soon extended to the subjects of mechan- ics, astronomy, and indeed, in a greater or less degree, to all branches of natural philosophy; so that, at the present time, all the varieties of physical phe- nomena, with which the higher branches of the science are conversant, are found to answer to vari- eties determinable by the algebraic analysis. ANALYSIS. 143 Two classes of quantities, and consequently two sets of symbols, quite distinct from each other, enter into this Calculus ; the one called Constants, which preserve a tixed or given value throughout the same discussion or investigation ; and the other called Variables, which 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 branch of mathematical science which examines, discusses, and develops the properties of the geometrical mag- nitudes by noting the changes that take place in the algebraic symbols which represent them, the laws of change being determined by an algebraic equation or formula. ANALYTICAL TRIGONOMETRY. 191. Analytical Trigonometry is that branch of Mathematics which treats of the general properties and relations of the sides and angles of Triangles, by means of Analysis. DIFFERENTIAL AND INTEGRAL CALCULUS. 192. In this branch of mathematical science, as in Analytical Geometry, two kinds of quantity are considered, viz. Variables and Constants ; and, con- 144 MATHEMATICS. sequently, two distinct sets of symbols are employed. The science consists of a series of processes which note the changes that take place in the values of the Variables. Those changes of value take place accord- ing to fixed laws established by algebraic formulas, and are indicated by certain marks drawn from the variable 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 general and minute laws which regulate its changes of value. 193. It will be observed, that Analytical Geom- etry and the Differential and Integral Calculus treat of quantity regarded under the same general aspect, viz. as subject to changes or variations in magnitude, according to laws indicated by algebraical formulas; and the quantities, whether variable or constant, are, in both cases, represented by the same algebraic sym- bols, viz. the constants by the first, and the varia- bles by the final letters of the alphabet. There is, however, this important difference: in Analytical Ge- ometry all the results are inferred from the 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. ANALYSIS. 145 194. Algebra, Analytical Geometry, the Differ- ential and Integral Calculus, extended into the Theo- ry of Yariations, make up the subject of analytical science, of which Algebra is the elementary branch. As the limits of this work do not permit us to dis- cuss the subject in full, we shall confine ourselves to Algebra, pointing out, occasionally, a few of the more obvious connections between it and the two other branches. SECTION VI ALGEBRA. 195. On an analysis of the subject of Algebra, we think it will appear that the subject itself pre- sents no serious difficulties, and that most of the embarrassment which is experienced by the pupil in gaining a knowledge of its principles, as well as in their applications, arises from not attending suffi- ciently to the language or signs of the thoughts which are combined in the reasonings. 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 significa- tion of the characters that stand for the quantities which are the subjects of the analysis; and also, of those signs which indicate the several operations to be performed on the quantities. 196. The Quantities which are the subjects of the algebraic analysis may be divided into two classes : those which are known or given, and ^those which are ALGEBRA. 147 unknown or sought. The known are uniformly repre- sented by the first letters of the alphabet, a, b, c, d, &c; and the unknown, by the final letters, a?, y, s, v, w, &c. 197. Quantity is susceptible of being increased or diminished and measured ; and there are six oper- ations which can be performed upon a quantity that will give results differing from the quantity itself, viz.: 1st. To add it to itself or to some other quantity; 2d. To subtract some other quantity from it; 3d. To multiply it by a number; 4th. To divide it; 5 th. To raise it to any power ; 6th. To extract a root of it. The cases in which the multiplier or divisor is 1, are of course excepted ; as also the cases where a root is to be extracted of 1, or 1 raised to any power. 198. The six signs which denote these operations are too well known to be repeated here. These, with the signs of equality and inequality, the letters of the alphabet and the figures which are employed, make up the elements of the algebraic language. The words and phrases of the algebraic, like those of every other language, are taken in connection with each other, and are not to be interpreted as sepa- rate and isolated symbols. 148 MATHEMATICS 199. The symbols of quantity are designed to represent quantity in general, whether abstract or concrete, whether known or unknown; and the signs which indicate the operations to be 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 which it is placed is to be added to some other quantity ; and the sign minus implies the existence of a minuend, from which the subtrahend is to be taken. One thing should be observed in regard to the signs which indicate the operations that are to be performed on quantities, viz. they 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 quan- tity. In Algebra, for example, the minus sign merely indicates that the quantity before which it is written is to be subtracted from some other quantity; and in Analytical Geometry, that the line before which it falls is estimated in a contrary direc- tion from that in which it would have been reck- oned, had it had the sign 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. The interpretation of the language of Algebra is the first 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 sig- ALGEBRA. 149 nifications and uses are as familiar as the sounds and combinations of the letters of the alphabet. 200. Beginning with the elements of the lan- guage, let any number or quantity be designated by the letter a, and let it be required to 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 regarding a as one #, or 1 a, and then observing that one a and one a, make two a's or 2 a : hence, a-\- a =2 a] and thus we place a figure before a letter to indicate how many times it is taken. Such figure is called a Coefficient. 201. The product of several numbers is indi- cated by the sign of multiplication, or by simply writing the letters which represent the numbers by the side of each other. Thus, a x ft x c x d xf, or aocdf, indicates the continued product of a, ft, c, d, and f\ and each letter is called a factor of the product: 150 MATHEMATICS hence, a factor of a product is one of the multipliers which produce it. Any figure, as 5, written before a product, as 5 a he df, is the coefficient of the product, and shows that the product is taken 5 times. 202. If the numbers represented by a, 5, c, d, and f were equal to each other, they would each be rep- resented by a single letter , and the product would then become axaxaxaxa = a 5 ; that is, we indicate the product of several equal fac- tors by simply writing the letter once and placing a figure above and a little at the right of it, to indicate how many times it is taken as a factor. The figure so written is called an exponent. Hence, an exponent is a simple form of expression, to point out how many equal factors are employed. 203. To denote that any quantity denoted by a, is to be raised to any power, as the fifth, for example, we merely write it with an exponent 5 ; thus, which is read, a to the fifth power. ALGEBRA. 151 204. The division of one quantity by another is indicated by simply writing the divisor below the divi- dend, after the manner of a fraction ; 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, , h -r tf, or, b | #, Cb indicates the division of h by a. 205. The extraction of a root is indicated by the sign sf. This sign, when used by itself, indicates the lowest root, viz. the square root. If any other root is to be extracted, as the third, fourth, fifth, &c, the figure marking the degree of the root is written above and at the left of the sign ; as, v denotes the cube root; V7~fourth root, &c. The figure so written, is called the Index of the root. We have thus given the very simple and general language by which we indicate each of the six opera- tions that may be performed on an algebraic quantity, and every process in Algebra involves one or other of these operations. 152 MATHEMATICS. MINUS SIGN. 206. The algebraic symbols are divided into two classes entirely distinct from each other, viz. the sym- bols which represent quantities, and the signs which denote operations. "We have seen that all the algebraic processes are comprised under addition, subtraction, multiplication, division, raising of powers, and the extraction of roots ; and it is plain, that the nature of a quantity is not at all changed by prefixing to it the sign which indicates either of these operations. The quantity denoted by the letter a, for example, is the same, in every respect, whatever sign may be pre- fixed 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 other root of it. The algebraic signs, therefore, must be regarded merely as indicating operations to be performed on quantity, and not as affecting the nature of the quantities to which they may be prefixed. "We say, indeed, that quantities are plus and minus, but this is an abbreviated language to express that they are to be added or subtracted. 207. In Algebra, as in Arithmetic and Geometry, all the principles of the science are deduced from the definitions and axioms; and the rules for performing ALGEBRA. 153 the operations are but directions framed in conformity to sncli principles. 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 subtracted from another quantity, we wish to discover the process of performing that subtraction, so as to deduce therefrom a general principle, from which we can frame a rule applicable to all similar cases. SUBTRACTION. 208. Let it be required, for example, to subtract from b the difference between a and c. Now, having written the letters, with their proper signs., the language of Algebra ex- presses that it is the difference only between a and c, which is to be taken from b; and if this difference were known, we could make the subtraction at once. But the nature and generality of the algebraic sym- bols, enable us to indicate operations, merely, and we cannot in general make reductions until we come to the final result. In what general way, therefore, can we indicate the true difference ? If we indicate the subtraction of a from b, we have b a ; but then, we have taken away too' much from b by the number of ba b a + o units in c, for it was not a, but the difference between a and o that was to be subtracted from 5. Having 7* 154 MATHEMATICS, taken away too much, the remainder is too small by c: hence, if o be added, the true remainder will be ex- pressed by b a + o. Now, by analyzing this result, we see that the sign of every term of the subtrahend has been changed; and what has been shown with, respect to these quantities is equally true of all others standing in the same relation : hence, we have the follow- ing general rule for the subtraction of algebraic quantities : Change the sign of every term of the subtrahend, or conceive it to he changed, and then unite the quan- tities as in addition. MULTIPLICATION. 209. Let us now consider the case of multiplica- tion, and let it be required to multiply a b by o. The algebraic language expresses that the difference between a and b is to be taken as many times as there are units in c. If we ab knew this difference, we could at once c perform the multiplication. But by what general process is it to be performed with- out finding that difference? If we take a, c times, the product will be ac : but as it was only 'the dif- ference between a and b, that was to be multiplied by c\ therefore, this product ac will be too great by acbc ALGEBRA. 155 b taken c times ; that is, the true product will be ex- pressed by acbc. Hence, we see, that, If a quantity having a plus sign be multiplied by another quantity having also a plus sign, the sign of the product will be plus ; and if a quantity hav- ing a minus sign be midtiplied by a quantity hav- ing a plus sign, the sign of the product will be minus. iJlO. Let us now take the most general case, viz. that in which it is required to multiply a b by cd. Let us again observe that the algebraic language denotes that a b is to be taken as many times as there are units in o d; and if these two dif- ferences were known, their prod- uct would at once form the prod- uct required. First: let us take a b as many times as there are units in o ; this product, from what has already been shown, is equal to acbc. But since the multiplier is not c, but cd, it follows that this product is too large by a b taken d times; that is, by adbd: hence, the first product diminished by this last, will give the true product. But, by the rule for subtrac- tion, this difference is found by changing the signs ab cd acbc ad + bd acbc ad -\-bd of the subtrahend, and then uniting all the terms 156 MATHEMATICS in addition : hence, the true product is expressed by acbcad + bd. By analyzing this result, and employing an ab- breviated language, we have the following general principle to which the signs conform in multiplica- tion, viz. : Plus multiplied by plus gives plus in the product ; plus multiplied by minus gives minus / minus multi- plied by plus gives minus f and minus multiplied by minus gives plus in the product. 211. The remark is often made by pupils that the above reasoning appears very satisfactory so long as the quantities are presented under the above form; but why w T ill & 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 prefixed to b and d, shows that in an algebraic sense they do not stand by themselves, but are connected with other quantities : and if they are not so connected, the minus sign makes no difference; for, it in no case affects the quantity, but merely points out a connection with other quantities. Besides, the prod- uct determined above, being independent of any par- ticular value attributed to the letters a, b, c, and d, must be of such a form as to be true for all values; and hence for the case in which a and c are both ALGEBRA. 157 equal to zero. Making this supposition, the prod- uct reduces to the form of + bd. The rules for the signs in division are readily de- duced from the definition of division, and the prin- ciples already laid down. ZERO AND INFINITY. 212. The terms zero and infinity have given 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 mathematical science, the precise ideas which these terms are designed to ex- press ; but we are unwilling to suppose that the ideas themselves are beyond the grasp of an ordinary intellect. The terms are used to designate the two limits of each of the quantities, Space, Number, and Time. 213. Assuming any two points in space, and join- ing them by a straight line, the distance between the points will be truly indicated by the length of this line, and this length may be expressed numerically by the number of times which the line contains a known unit. If, now, the points are made to approach each other, the length of the line will diminish as the 158 MATHEMATICS. points come nearer and nearer together, until at length, when the two points become one, the length of the line will disappear, having attained its limits which is called zero. If, on the contrary, the points recede from each otner, the length of the line joining them will con- tinually increase; but so long as the length of the line can be expressed in terms of a known unit of 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. 214. Assuming one as the unit of number, and admitting the self-evident truth that it may be in- creased .or diminished, we shall have no difficulty in understanding the import of the terms zero and in- finity, as applied to number. For, if we suppose the unit one to be continually diminished, by division or otherwise, the fractional units thus arising will be less and less; and in proportion as we continue the di- visions, they will continue to diminish. Now, the limit or boundary to which these very small fractions 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. ALGEBRA. 159 If, oiTtlie other hand, we suppose a number to be continually increased, the relation of this 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 infinite; but when the unit one forms no appreciable part of the number, the term infinite is used to express that state of value, or, rather, that limit of value. 215. The terms zero and infinity are there- fore employed, to designate the limits to 'which de- creasing and increasing quantities may be made to approach nearer than any assignable quantity; but these limits cannot be compared, in respect to magni- tude, with any known standard, so as to give a finite ratio. 216. It may, perhaps, appear somewhat paradoxi- cal, that zero and infinity should be defined as " the limits of number and space" when they are in them- selves not measurable. But a limit is that " which sets bounds to, or circumscribes;" and as all finite space and finite number (and such only are implied by the terms Space and Number), are contained between zero and infinity, we employ these terms to designate the limits of Number and Space. 160 MATHEMATICS. OF THE EQUATION. 217. We have seen that all deductive reason ins: involves certain processes of comparison, and that the syllogism is the formula to which those processes may be reduced. It has also been stated that if two quan- tities be compared together, there will necessarily result the condition of equality or inequality. The equation is an analytical formula for expressing equality. 218. The subject of equations is divided into two parts. The first, consists in finding the equation; that is, in the process of expressing the relations existing be- tween the quantities considered, by means of the alge- braic symbols and formula. This is called the State- ment of the proposition. The second is purely deduc- tive, and consists, in Algebra, in what is called the solution of the equation, or finding the value of the unknown quantity : and in the other branches of analy- sis, it consists in the discussion of the equation ; that is, in the drawing out from the equation every thing which it is capable of expressing. 219. Making the statement, or finding the equa- tion, is merely analyzing the problem, and expressing its elements and their relations in the language of anal- ysis. It is, in fruth, collating the facts, noting their ALGEBRA. 161 bearing and connection, and inferring some general law or principle which leads to the formation of an equation. The condition of equality between two quantities is expressed by the sign of equality, which is placed be- tween them. The quantity on the left of the sign of equality is called the first member, and that on the right, the second member of the equation. The first member corresponds to the subject of a proposition ; the sign of equality is copula and part of the predi- cate, signifying, is equal to. Hence, an equation is merely a proposition expressed ^algebraically, in which equality is predicated of one quantity as compared with another. It is the great formula of analysis. 220. We have seen that every quantity is either abstract or denominate ; hence, an equation, which is a general formula for expressing equality, must be either abstract or denominate. An abstract equation expresses merely the relation of equality between two abstract quantities : thus. a + h = x, 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 ex- 162 MATHEMATICS. presses that the sum of a and h is equal to a?, and is true, equally, of all quantities. But if we assign a denominate 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 equa- tion will become denominate. 221. "We have seen that there are six operations which may be performed on an algebraic quantity. We assume, as an axiom, that if the same operation, under either of these processes, be performed on both members of an equation, the equality of the mem- bers will not be changed. Hence, we have the six following AXIOMS. 1. If equal quantities be added to both members of an equation, the equality of the members will not be destroyed. 2. If equal quantities be substracted from both members of an equation, the equality will not be destroyed. 3. If both members of an equation be multiplied by the same number, the equality will not be de- stroyed. 4. If both members of an equation be divided by ALGEBRA. 163 the same number, the equality will not be de- stroyed. 5. If both members of an equation be raised to the same power, the equality of the members will not be destroyed. 6. If the same root of both members of an equa- tion be extracted, the equality of the members 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. 222. The term Equality, in Geometry, expresses that two magnitudes have equal measures; that is, that they contain the same unit an equal number of times. It has the same signification in Algebra. G-ENERAL REMARKS. 223. We have thus pointed out some of the marked characteristics of analysis. In Algebra, the elementary branch, the quantities, about 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 164 MATHEMATICS. relations, and of finding the values of the unknown quantities, in terras of the known. In the other branches of analysis, the quantities considered are divided into two general classes, Con- stant and Variable; the former preserving fixed val- ues throughout the same process 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 properties applicable to all quantities. 224. The correspondence between the processes of reasoning, as exhibited in the subject of general logic, and those which are employed in mathemat- ical 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 subject to which the reasoning is applied, does not at all change the nature of the process, or materially vary its form. 225. We shall not pursue the subject of analysis any further; for, it would be foreign to the pur- poses of the present work to attempt more than to point out the general features and characteristics of the different branches of mathematical science. We have aimed only to present the subjects about which ALGEBRA. 165 the science is conversant, to explain the peculiari- ties of the language, the nature of the reasoning pro- cesses employed, and of the connecting links of that golden chain which binds together all the parts, forming an harmonious whole. SUGGESTIONS FOR THOSE WHO TEACH ALGEBRA. 1. Be careful to explain that the letters employed, are the mere symbols of quantity. That of, and in themselves, they have no meaning or signification whatever, but are used merely as the signs or rep- resentatives of such quantities as they may be em- ployed to denote. 2. Be careful to explain that the signs which are used are employed merely for the purpose of indica- ting the six operations which may be performed on quantity ; and that they indicate operations merely, without at all affecting the nature of the quantities before which they are placed. 3. Explain that the letters and signs are the ele- ments of the algebraic language, and that the sym- bolical language itself arises from the combination of these elements. 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 166 MATHEMATICS. of Algebra ; and that the interpretation of an alge- braic formula is merely translating its various sig- nifications into common language. 5. Let the language of Algebra be carefully studied, so that its construction and significations may be clearly apprehended. 6. Let the difference between a coefficient and an exponent be carefully noted, and the office of each often explained ; and illustrate frequently the sig- nification of the language by attributing numerical values to letters in various algebraic expressions. T. Point out often the characteristics of similar and dissimilar quantities, and explain which may be incorporated, and which cannot. 8. Explain the power of the minus sign, as shown in the four ground-rules ; but very particularly, as it is illustrated in subtraction and multiplication. 9. Point out and illustrate the correspondence be- tween the four ground-rules of Arithmetic and Algebra ; and impress the fact, that their differ- ences, wherever they appear, arise merely from dif- ferences in notation and language : the principles which govern the operations being the same in both. 10. Explain with great minuteness and particular- ity all the characteristic properties of the equation : the manner of forming it ; the different kinds of quantity which enter into its composition ; its exam- ALGEBRA. 167 ination or discussion ; and the different methods of elimination. 11. In the equation of the second degree, be care- ful to dwell on the four forms which embrace all the cases, and illustrate by many examples that every equation of the second degree may be re- duced to one or other of them. Explain very par- ticularly the meaning of the term root; and then show, why every equation of the first degree has one, and every equation of the second degree two. Dwell on the properties of these roots in the equa- tion of the second degree. Show why their sum, in all the forms, is equal to the coefficient of the second term, taken with a contrary sign ; and why their product is equal to the absolute term with a contrary sign. Explain when and w T hy the roots are imaginary. 12. In fine, remember that every operation 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 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 sub- lime science which discovers the laws of nature in 168 MATHEMATICS. their most 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. RETURN CIRCULATION DEPARTMENT TO 202 Main Library LOAN PERIOD 1 ~ HOME USE 2 3 4 5 ( 5 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Renewals and Recharges may be made 4 days prior to the due date. Books may be Renewed by calling 642-3405. DUE AS STAMPED BELOW SEP 18 1987 1 ^d^filt* 7 JUL 02 1988 - UNIVERSITY OF CALIFORNIA, BERKELEY FORM NO. DD6, BERKELEY, CA 94720 $ U. BERKELEY LIBRARIES B0030nb 2a