Refolution of a problem, bow to take away an aliquot part from a given line, and the resolution of this problem fuppofing the doctrine of fimilitude, could not be given but in the IX. Propofition of the VI. Book.

VI.

Magnitudes which have the fame ratio, are called proportionals.

When four magnitudes A,B,C,D are proportional, it is ufually expreft thus, A: BC: D and in words, the first is to the second as the third to the fourtb. VII.

When of the equimultiples of four magnitudes (taken as in the 5th definition) the multiple of the first is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the fecond a greater ratio than the third magnitude has to the fourth, and on the contrary, the third is faid to have to the fourth a lefs ratio than the first has to the fecond.

§. 1. Such are the ratios 3 2 & 11 : 9 for if the antecedents be multiplied by 9, and the confequents by 13, there will refult 27: 26; 99: 117.

32; 11: 9

9 13 9 13

27:26; 99: 117

Where the correspondence of the multiples does not bold, the firft antecedent 21 being greater than its confequent 26 whilst the second antecedent 99 is less than its confequent 117.

§. 2. To difcover by infpection the inequality of two ratios A: B & C D by this character of the non correfpondence of multiples, it fuffices to chufe for multiples, the two terms of one of the two ratios, for Ex. C: D, and to multiply the antecedents A & C by the confequent D of this ratio; and the two confequents B & D by the antecedent C of the fame ratio, in this manner.

:

A B C D
A:. ;
D C; D: C

AD: BC; CD: D. C

35; 7: 9
9 7 9 7

27:35; 63; 63

Which being done, the two products C.D & D.C will be found equal, whilft the two others A.D & B.Ċ are unequal, and in particular, if the multiple of one of the antecedents be greater than that of its confequent, whilft the multiple of the other is equal to its, then the terms of the leffer ratio have been chofen for multipliers. On the contrary, if the multiple of one of the antecedents be less than that of its confequent, whilst the multiple of the other is equal to its, then the terms of the greater ratio bave been chofen for multipliers.

VIII.

Analogy or proportion, is the fimilitude of ratios.

5.)

As a fign and character of proportionals bas been already given (in Def. 5. this is a fuperfluous definition, a remark of fome fcholiaft fufled into the text which interrupts the coberence of Euclid's genuine definitions.

IX.

Proportion confifts in three terms at least.

§. 1. Proportion confifting in the equality of two ratios, and each ratio baving two terms, in a proportion there are four terms, of which the first and fourth are called the extreames, and the fecond and third the means, thofe four terms are confidered as only three, when the confequent of the firft ratio at the fame time bolds the place of the antecedent of the fecond ratio: it is for this reason, that proportions are diftinguifbed into difcrete, and continued. A proportion is difcrete when the two means are unequal, and it is called continued when these Same terms are equal, thus this proportion 2 : 4 = 5: 10 is difcrete because the two mean terms 4 & 5 are unequal, on the contrary, the proportion 2:44 8 is a continued proportion on account of the equality of the mean terms 4

فوع

4.

§. 2. A feries of magnitudes in continued proportion, forms a geometrical progreffion, fuch are the numbers 1, 2, 4, 8, 16, 32, 64, &c.

X.

When three magnitudes are proportional the first is said to have to the third the duplicate ratio of that which it has to the second.

XI.

When four magnitudes are continual proportionals, the first is faid to have to the fourth the triplicate ratio of that which it has to the fecond, and so on quadruplicate, &c. increafing the denomination still by unity in any number of proportionals.

XII.

In proportionals, the antecedent terms are called Homologous to one another, as alfo the confequents to one another.

XIII.

Proportion is faid to be alternate when the antecedent of the first ratio is compared with the antecedent of the fecond, and the confequent of the first ratio with the confequent of the fecond.

If A : B = C: D then by alternation. {

4:5 = 16: 20

A: C = B: D
4165: 20

When the proportion is difpofed after this manner, it is faid to be done by permutation or alternately, permutando or alternando.

XIV.

But when the confequents are changed into antecedents, and the antecedents into confequents in the fame order, it is faid that the comparison of the terms is made by inverfion or invertendo.

A: BC: D therefore invertendo. B: A=D: C

394: 12

XV.

=

93 12:4

But the comparison is made by compofition or componendo, when the fum of the confequents and antecedents is compared with their respective confequents.

ABCDS therefore (A + B ; B = C+D: D 39 4:12 componendo 3 +9:9=4+12:12

XVI.

The comparison is made by divifion of ratio, or dividendo when the excess of the antecedent above its confequent, is compared with its confequent.

If A B = C: D
9:312: 4

dividendo. { SA-B:B=C-D:D

XVII.

9-33=12-44

The comparison is made by the converfion of ratio, or convertendo, when the antecedent is compared to the excefs of the antecedent above its confequent. If A: B = C: D 9:312: 4

therefore {A: A—BC: C-D convertendo. 9: 9—3 = 12 : 12-4 XVIII.

A conclufion is drawn from equality of ratio or ex æquo, when comparing two feries of magnitudes of the fame number, fuch that the ratios of the first be equal to the ratios of the fecond, each to each, (whether the comparison be made in the fame order or in an inverted one), it is concluded that the extreames of the two feries are in proportion.

The fenfe of this definition is as follows, if A, B, C, D be a feries of four magnitudes, and a, b, c, d a series of four other magnitudes, fuch that

A: Ba: b

A: B = c : d

(C: D=a: b

B: C =b:c or in an inverted order. B: C =b:c
C D = c: d.

In the one or the other cafe it is allowed to infer ex æquo, when the ratio of the extreames A: D of the I. feries is equal to the ratio of the extreames a : d of the II. feries; or that A: D =≈ a :`d.

The equality of ratio is called ordinate ratio, when the ratio of the first feries are equal to the ratios of the second series each to each in the fame direct order.

Here the ratios are equal each to each in the fame direct order, because the firft magnitude is to the fecond of the firft rank, as the first to the fecond of the other rank, and as the fecond is to the third of the first rank, fo is the fecond to the third of the other, and fo on in order. on in order. If therefore it is inferred that the extreames are proportional, or that A: Da: d. the inference is faid to be made from direct equality, or ex æquo ordinate.

XX.

On the contrary, equality of ratio is called inverted or perturbate analogy, in the second case, that is when the ratios of the first series are equa to those o the second series each to each, taking those last in an inverted order.

§. 1. Let the two series of magnitudes be.

A, B, C, D
a, b, c, d

A B C d where it is fuppofed B C b: c C: Da: b

=

Here the ratios of the I. feries are equal to the ratios of the Il. feries each to each, but in an inverted order, that is the firft magnitude is to the fecond of the first rank, as the laft but one is to the laft of the fecond rank, and as the Jecond is to the third of the firft rank, fo is the last but two to the laft but one of the fecond rank; and as the third is to the fourth of the first rank, fo is the tbird from the laft to the last but two of the fecond rank, and fo in a cross order. If therefore it be inferred that A: Dad.

This inference is faid to be made ex æquo perturbate.

§. 2. Beginners may easily diftinguish the cafe of direct equality from that of perturbate equality, if they remember that when two terms are common to two proportions, and that they occupy indifferently either the firft and third, or the fecond and fourth place, that it is always the cafe of direct equality ; For Example.

Here are always two proportions which have in common the two terms B&b occupying the firft and third, or the fecond and fourth places; the two other terms A & C are proportional to the two others a &c taking them in the fame

order.

§. 3. On the contrary when the two terms which are common to the two proportions, are either the means or the extreames, it is the cafe of perturbate equality, for example

In those three cafes the terms B&b which are common to the two proportions, are either the extremes or the means; confequently the other terms are in proportion, fo that the two terms, which arise from the same proportion A&C

or a & c remain extreams or means.

These are the denominations given to the different ways of concluding by analogy, Euclid now proceeds to demonftrate that they are just.