The pro 595. If we extend our inquiry to the determination of the ductof three meaning of the product abc of three factors a, b, c, which sevepresenting rally represent geometrical straight lines, we symbols re lines, re presents the contain. C g e 모 B shall find that it may correctly represent the solid paral- volume or solid content of the rectangular G lelopipedon which they parallelopipedon, of which a, b and c are three adjacent edges: for if AB (a), AC (b), AD (c), the three adjacent edges of such a parallelopipedon, (any one of which is perpendicular to the other two, or to the plane which passes through them,) are expressible by integral numbers, and are severally divided into equal linear units, (such as Ab, or Ac, or Ad), and if planes parallel to the bounding planes of the parallelopipedon were made to pass through the several points of division, the solid ABFDGCE would be divided into cubes equal to each other (and to becgdf), constructed upon a linear unit, the number of which will be equal to the product of the numbers which express the number of linear units in each edge respectively: the requisite arithmetical condition is therefore satisfied, it being merely necessary to keep in mind that the units in the product are equal cubes, whilst those in the factors are equal lines. Signs of the product of three fac tors, which may be positive or negative. This solid is of the same magnitude whatever be the order in which the symbols or the edges which they represent, are taken and the product abc denotes a rectangular and not an oblique parallelopipedon, inasmuch as the latter solid involves the values of the angles which the edges make with each other, and which its definition does not determine. 596. There are only two signs of the product of three factors, though they may arise from eight different combinations of the signs of the component factors; they are as follows: D pretation. These eight products would correspond to eight different, Their interthough equal and similar rectangular parallelopipedons, having a common angle (4), and constructed upon edges which form severally one of each of the pairs of lines represented by a (AB) and -a - a (Ab), +b (AC) and − b (Ac), +c (AD) and −c(Ad). Those pairs of solids, which touch by a common plane, and which have therefore two edges in common, have different signs: those pairs of solids which have one edge B only in common, have the same sign: whilst those pairs which have only one point in common, and all whose three edges have therefore different signs, have also different signs. four or bols. 597. We should fail in attempting to give the interpretation Products of of the meaning of the product of four or more symbols repre- more symsenting lines, or of two or more symbols representing areas, or of any other combinations of symbols representing lines, areas, or solids, which exceed three dimensions, inasmuch as there is no prototype in Geometry with which such products can be compared in other words, the existence of such products is possible in symbols only. bolical 598. We have assigned an interpretation to the products Geometry ab and abc, when the symbols, which they involve, represent considered may be geometrical lines, in conformity with the general principle which as a symconnects Symbolical with Arithmetical algebra, and which assumes science. that when the symbols are replaced by numbers, such products degenerate into ordinary arithmetical products: if we may suppose, therefore, lines to represent numbers, (and there is no relation of magnitude which they may not represent,) they may equally represent any concrete magnitudes whatsoever, of which these numbers are the representatives: it will follow, therefore, that if a, b and c are represented by lines, the rectangle contained by the lines a and b, and the rectangular parallelopipedon constructed upon the lines a, b and c, may represent any specific magnitudes, which ab and abc may represent, when a, b and c are replaced by numbers. We shall thus give to Geometry the character of a symbolical science. Space described in uniform 599. Thus, if v represents the uniform velocity of a body's motion, and t the time during which it is continued, the product motion may vt will represent the space over which the body has moved be represented by a in the time t: and if we assume one line to represent v, and rectangular another line to represent t, the rectangle contained by them would represent the space described equally with the symbolical or numerical product vt. area. Linear representations of 600. When, however, lines are assumed to represent quantities like velocity (v) and time (t), which are different in their nature, and therefore admit of no comparison with each other in perfectly respect of magnitude, the first assumption of them must be perarbitrary. fectly arbitrary: thus, if v denoting a certain velocity, be repre units of different kinds Numerical units per trary. sented by an assumed line, v' denoting any other velocity, would be represented by another line bearing the same proportion to the former, that v' bears to v: and in a similar manner, one line may represent a time t, and another line any other time t', if they bear to each other the proportion of t to t': but the magnitudes v and t admit of no comparison with each other, and therefore the line which represents an assigned magnitude of one of them can bear no determinate relation to the line which represents an assigned magnitude of the other: in other words, the lines, which severally represent their units, may be assumed at pleasure. 601. The same remark applies, and for the same reasons, fectly arbi- to the representation of essentially different quantities by means of numbers, the values of their primary units being perfectly arbitrary: thus the unit of time may be a second, a minute, an hour, &c. whilst the unit of space or velocity, (for one is the measure of the other) may be a foot, a yard, a mile, &c.: thus, if the units be assumed to denote severally a second of time and a foot in space, we may speak of a velocity denoted by 1, 2, 3, 10 or 20, being such as would cause a body to move uniformly over 1, 2, 3, 10 or 20 feet of space in one second, twice those spaces in 2 seconds, three times those spaces in 3 seconds, and therefore through a space which would be denoted by vt, if the body moved with a velocity equal to v, during any number of seconds denoted by t: and if we now pass from arithmetic to geometry, we may assume a line to represent a second of time, whilst an equal or any other line represents a foot in space or a velocity of one foot: such primary units being once assumed, all other values of those magnitudes will be represented by lines bearing a proper relation to them. 602. The meaning of algebraical products, when the factors Algebraical are any assigned quantities, being once determined, we expequotients. rience no difficulty in interpreting the meaning of algebraical quotients, when the dividend and divisor are assigned both in representation and value: the general principle of such interpretations being, that "the operation of Division is in all cases the inverse of that of Multiplication:" in other words, the quotient or result of the division must be such a quantity, that, when multiplied into the divisor, it will produce the dividend: we will mention a few cases. 603. If the dividend and divisor be both of them abstract Examples numbers, the quotient is either an abstract number or a numerical fraction (Art. 92). If the dividend be concrete and the divisor an abstract number or numerical fraction, the quotient is a concrete quantity of the same kind with the dividend (Art. 590). If the dividend and divisor be concrete quantities of the same kind, the quotient is an abstract number or a numerical fraction (Art. 590). If the dividend be an area and the divisor a line, the quotient is a line which contains, with the divisor, a rectangular area equal to the dividend, (Art. 592). If the dividend be a solid and the divisor a line, the quotient is the rectangular base of an equal rectangular parallelopipedon, of which the divisor is the third edge (Art. 595). If the dividend be a space described and the divisor the uniform velocity with which it is described, the quotient is the time of describing it (Art. 599). It is not necessary, however, to multiply examples of such interpretations of the meaning of quotients, when the principle which connects them with their corresponding products admits of such easy and immediate application. of their interpretation. CHAPTER XIII. ON THE DETERMINATION OF THE HIGHEST COMMON DIVISORS Explana tion of the common divisors and lowest common AND THE LOWEST COMMON MULTIPLES OF TWO OR MORE 604. THE determination of the highest common divisors meaning of and the lowest common multiples of two or more algebraical highest expressions will be required for the reduction of fractions to their most simple equivalent forms, in the same manner that the processes for finding the greatest common measure and the multiples least common multiple of two or more numbers are involved of algebraical expres- in the corresponding reductions of numerical fractions (Arts. 98 and 116): we use the terms highest and lowest, with respect to the dimensions of the symbol of reference (Art. 576.) according to whose powers the terms of those expressions, whose common divisors or multiples are required, are arranged: the terms greatest and least ceasing to be applicable, in the case of expressions whose symbols are indeterminate in value. sions. whose highest Two alge- 605. We shall begin by considering the process for disbraical expressions covering the common divisor of two algebraical expressions only, and we shall arrange them as the numerator and denominator of a fraction, which it is proposed to reduce to its most common divisor is simple form: for we have already shewn (Arts. 75 and 578.) required, may be as- that any factor common to the numerator and denominator of form a frac- a fraction may be obliterated without altering its signification or tion which value. There are several steps in this process, which it will be duced to its convenient to notice in their order. sumed to is to be re most simple form. Detection of simple numerical factors. Examples. 606. In the first place, there may exist a common numerical divisor of the coefficients of all the terms of the numerator and denominator, which may be discovered by inspection, or by the common arithmetical rule (Art. 98.) for that purpose: thus, 3 is a divisor of every term of the numerator and denominator of the fraction 9x+15y 12x2 + 21 y3' |