BOOK V. DEFINITIONS.] 



MATHEMATICS. PLANE GEOMETRY. 



579 



measurable, or beyond the powers of numerical repre- 

 sentation. He has executed his difficult task with con- 

 summate ability ; for, as Dr. Barrow remarks, " there is 

 nothing, in the whole body of the Elements, of a more 

 subtile invention nothing more solidly established, and 

 more accurately handled, than the doctrine of pro- 

 portionals." 



It is on account of the subtleties here adverted to, and 

 which are of too refined a character for the generality 

 of young students to comprehend, that we have resolved 

 to replace the fifth book by the following treatise. We 

 cannot promise that you will find the study of it easy; 

 but it will certainly be much less difficult than the cor- 

 responding portion of Euclid's work ; and you will enter 

 upon it with considerable advantage, if you have read 

 Arithmetical and Geometrical Progression in the chapter 

 on Algebra. 



DEFINITIONS. 



I. Of two unequal magnitudes, of the same kind, the 

 greater is said to contain the less as many times as there 

 are parts in the greater equal to the less. 



TbU definition is intended to convey the sense in which the word 

 contain is to be understood in what follows. You will carefully 

 observe that the term is not restricted by any condition as to 

 whether or not the greater quantity leave any remainder after 

 taking; the less from it as often as possible. Thus 8 is said to con- 

 tain 2 as often as 9 contains it, though there Is a remainder in the 

 latter case, and no remainder in the former. 



II. One magnitude is said to be a multiple of another 

 when it is equal to a certain number of times that other 

 exactly. And the less of the two is in this case said to 

 be a tub-multiple of the greater, or a measure of the 

 greater. 



Thus 8 is a multiple of 2, because 8 la equal to a certain number of 

 time* 2 exactly; namely, four times 2. And 2 is a sub-multiple of 

 8, or, as we should say in arithmetic, > factor of 8. Remember that 

 a sub-multiple, or mraiure of any magnitude, is a smaller magni- 

 tude of the same kind which measures the former (the multiple) 

 exactly, without leaving any remainder. 2 is not a sub-multiple or 

 measure of 9 ; which is the same as saying, that 9 i* not a multiple 

 of 2. 



III. Magnitudes two or more which have a common 

 mature that is, which are multiples of some other mag- 

 nitude are said to be commensurable. But if it be im- 

 possible that any such common measure can exist, then 

 the magnitudes are said to be incommensurable. 



All seta of>batract numbers, and of concrete quantities, like in kind, 

 that can be accurately denoted by numbers, are commensurable ; 

 that is, they hare a common measure. The expression, " common 

 measure," as employed in arithmetic and algebra, is not synony- 

 mous with the geometrical meaning. In arithmetic we should say 

 that the pairs of numbers 3, 7 ; 4. 11 ; 5, 12, &c., have no common 

 measure ; but even here a qualifying exception is always tacitly 

 made: it is this namely, except unit. All whole numbers contain 

 1 an exact number of times ; though it is agreed in arithmetic that 

 1 shall not be recognised as a common measure. Geometry makes 

 no such exception : whatever quantity is contained in another, an 

 exact number of times, is a measure of that other. In like manner, 

 34, 7J, would not be regarded as having a common measure in 

 arithmetic ; yet as the first number contains J exactly 14 times, and 

 the second contains J exactly 29 times, i is a common measure of 

 the two numbers according to the above definition of the term. And 

 whatever numbers be compared together, whether they be whole or 

 fractional, it will be found that there always exist* some smaller 

 number either whole or fractional that will exactly measure 

 both. Suppose, for instance, the proposed numbers, when brought 

 to a common denominator, have (say) 12 for the common denomi- 

 nator, then each denotes so many twelfth* : that Is, ^ is a common 

 measure of both. We do not say anything here about such expres- 

 sions as y^2, y/3, \/7, & c * i geometrical rigour forbids our calling 

 what these symbols stand for, definite numbers, as they involve an 

 endless series of fractions or decimals, and can only be valued ap- 

 proximately ; yet they may be accurately represented by Imet, as 

 1 has himself shown in a Book not now read (Book X.) We 

 could not speak of any number being contained m /2 a certain 

 number of times exactly, because y/ 2 itself i* not determinate 

 exactly. 



IV. E'/uimitltiples, or like multiples, of two or more 

 magnitudes, are those larger magnitudes which contain 

 those of which they are multiples each of each the 

 tame number of times. 



For intmce, the numbers 8 and 12 are equimultiples pf 2 and 3; for 



the former contain these, respectively, the same number of times 



namely, fvur limes. In like manner, 7$ and 10 are equimultiples 

 of 2j and 3J ; for the former numbers contain these, respectively, 

 nines. 



V. And like tub-multiples are those which are con- 



tained in their respective multiples the same number of 

 times. 



Thus, in the instances just adduced, 2 and 3 are like sub-multiples 

 or like measures of 8 and 12 ; 2$ and 3 are like sub-multioles or 

 measures of 7$ and 10. 



VI. Four magnitudes are said to be proportionals, or 

 to form a proportion, when the first cannot be contained 

 in any multiple of the second, ofteuer than the third is 

 contained in a like multiple of the fourth, nor the third 

 in any multiple of the fourth, oftener than the first in a 

 like multiple of the second. 



The first and third of four such magnitudes are called antecedents, 

 and the second and fourth their consequents. The definition 

 affirms that an antecedent must be continued in each multiple of 

 its consequent as often as the other antecedent is contained in a 

 like multiple of its consequent, but not oftener. 



VII. When four magnitudes are in proportion, the 

 first antecedent is said to have the same ratio to its con- 

 sequent that the second antecedent has to its conse- 

 quent. 



This term ratio has been the source of very considerable embarrass- 

 ment to Geometers since the time of Euclid, and has been pro- 

 ductive of much metaphysical disquisition and controversy. We 

 think that on this subject, as well as in reference to the theory of 

 parallel lines, mathematicians too often overlook the fact that the 

 fundamental notions of geometry really exist in the mind anterior 

 to, and independently of, the definitions of the science. These, for 

 the most part, do not originate those notions, but only give to 

 them the necessary degree of clearness and precision. If one pair 

 of magnitudes be submitted to our contemplation, and then another 

 pair be brought into comparison with them, as well as with eaeh 

 other, the mind is at once capable of forming a notion as to whether 

 the relative magnitudes of the individuals of the first pair IK UIL' 

 same or not as the relative magnitudes of those of the second pair : 

 the absolute magnitudes of the individuals of one pair may be very 

 different from the absolute magnitudes of those of the other pair ; 

 yet the former two may have the same relation to one another, as 

 to magnitude, as the latter two ; and the mind is quite capable of 

 recognising and understanding this sameness of relation, or of 

 ratio, aa it is called above, before any name is given to the concep- 

 tion. It is this equality of ratios of two magnitudes brought into 

 comparison with other two, that renders the four proportionals. 



That the foregoing definition of proportion ( Def. VI. J includes nume- 

 rical proportion, in Arithmetic, will be obvious upon a little consi- 

 deration. Proportion limited to numbers may be defined thus : 



Four numbers are proportionals when the first is contained exactly, 

 as often in some multiple (any one multiple being sufficient) of the 

 second, as the third is contained in a like multiple of the fourth. 



It is plain that if this condition have place, the four numbers must 

 be proportionals according to the common arithmetic-. 1 notion ; tor 

 it follows of necessity, that the quotient of the second by the first 

 must then be the same as that of the fourth by the tbiid, and con- 

 sequently that the quotient of the first by the second must be tilt; 

 same as that of the third by the fourth.* 



Definition VI., above, is, however, free from the restriction implied in 

 the term exactly, which is introduced here into the particular <.,- 

 of it, applying exclusively to numerical proportion ; since the 

 general form of the definition admits of such restrictive quali- 

 fication when numbers only are concerned ; iiicomuiensurables 

 being then excluded. 



Four magnitudes not fulfilling the conditions of Definition VI. would 

 evidently violate even this arithmetical condition of proportion. 

 There can be no such things as proportionals out of the restrictions 

 of the former definition ; w> that all proportionals, whether among 

 commensurables or incommensurable*, must be included iu the 

 general Definition VI. 



VIII. The first and last of four proportionals are 

 called the extremes, and the two intermediate ones the 

 meant. 



IX. The magnitudes themselves are called the terms 

 of the proportion ; and those are called homologous or 

 like terms which have the same name ; the antecedents 

 forming one pair of homologous terms, aud the conse- 

 quents another pair. 



X. Magnitudes, more than two, are said to form a 

 continued proportion when each consequent in succession 

 is taken for the antecedent of the term next following. 



Thus, if A is to B, as B is to C, as C is to D, &c. ; then A, B, C, &c , 

 arc in continued proportion. 



XI. If the continued proportionals be but three in 

 number, the middle one is called the mean term, and the 

 others the extremes. 



I. Equimultiples of the same magnitude, or of equal 

 magnitudes, are equal to one another ; so also are equi- 

 Bub-multiples. 



II. A multiple of a greater magnitude exceeds a like 

 multiple of a less ; and a sub-multiple of a greater ex- 

 ceeds a like sub-multiple of a 1,3. 



Sec Arithmetic, p. UU. 



