PROP. XII. THEOR.

If, in a proportion, an antecedent be a multiple or submultiple of its consequent, the other antecedent will be a like multiple or submultiple of its consequent.

In the proportion A: B:: C: D let an antecedent as A be a multiple, mB, of its consequent B, then will C be a like multiple, mD, of D.

Take mD the same multiple of D that mB or A is of B

Then (prop. vi. cor.) A: mB :: C: mD.

But A= mB, therefore CmD (prop. vii.); that is, C is the same multiple of D that A is of B.

Again, let one of the antecedents A be a submultiple of its consequent B; then will C be a like submultiple of D.

For by inversion (prop. viii.) B: A:: D: C; therefore, as just shown, D is the same multiple of C that B is of A: in other words, C is the same submultiple of D that A is of B.

COR. Hence when the terms are homogeneous, if one antecedent be a multiple or submultiple of the other, the consequent of the former will be a like multiple or submultiple of the consequent of the latter, (prop. x.)

PROP. XIII. THEOR.

In a proportion, the sum of an antecedent and its consequent is to either term, as the sum of the other antecedent and consequent is to the like term.*

* It is remarkable how few of the commentators on Euclid have a correct notion of geometrical proportion. In the notes to the octavo work on geometry, by the Editor of the present treatise, some of the

Let the proportion be A: B :: C: D.

Then an antecedent A cannot be contained oftener in any multiple of its consequent B, than C is contained in a like multiple of D (def. 7.) Therefore A cannot be contained oftener in any multiple of A+B than C is contained in a like multiple of C + D.

Also C cannot be contained in a multiple of C+ D oftener than A is contained in a like multiple of A + B, for if it could, then C would be contained in the multiple of D oftener than A in the same multiple of B, which is impossible. Hence

A: A B C C+ D.

Again, by prop. viii.

BAD C;

more specious fallacies in the writings of modern Geometers upon this subject, are fully examined. We may observe here, in addition, that the present proposition which corresponds to the xviii. in the Elements, has been quite misapprehended by Mr. Keith, in his edition of Euclid. He proposes to replace Euclid's demonstration which is very prolix by a new and shorter proof, which he seems to regard as equally general. It is founded, however, upon two propositions, both of which apply only to homogeneous quantities, and which therefore cannot be admitted here, where one antecedent and its consequent may be any like quantities whatever, and the other antecedent and consequent also any like quantities whatever. It is probable that Keith adopted this erroneous demonstration from Austin, who offered it in his "Examination of Euclid," "as perhaps the best which can be given!" Another author, in a recent edition of the Elements, which has passed through three or four impressions, has in one place substituted a new demonstration for that of Simson (in proposition D) and has fallen into the common error, like Keith, of employing alternation to quantities whose antecedents are heterogenevus. We advert to these inconclusive demonstrations for the purpose of warning the student against incautiously giving his assent to general conclusions of which only particular cases are implied in the premises;-a fallacy of very frequent occurrence in writings on geometrical proportion. See the notes to Young's Elements of Geometry." 8vo. 1827.

AB: B: C + D : D.*

COR. When the terms are homogeneous, then (prop. x.)

In a proportion, the difference between an antecedent and its consequent is to either term as the difference between the other antecedent and consequent to the like term.

In the proportion A: B:: C: D take either antecedent and its consequent, as A and B ; and let first B > A, and consequently (prop. vii.) D > C.

Take any equimultiples of BA and D-C

A

B-A mB-MA

C

D-C mD-mC.

Then since (def. 7) A cannot be contained oftener in mB than C is contained in mD, and as A is contained in mA the same number of times as C is contained in mC, viz. m times; therefore A cannot be contained in mB-mA oftener than C is contained in mDmC.

In like manner is it shown that C cannot be contained in mD -mC oftener than A is contained in mB - mA and

This inference from the original proportion is called, by Euclid and the carly Geometers, the inference "by composition."

mB-mA, mD-mC are any equimultiples of B-A Hence (def. 7,)

and D.

Consequently (prop. viii.)

B A: A DC: C

Again, let A > B, and consequently C > D. Then (prop. viii.) B: A :: D: C, therefore from what has just been proved

ABB :: CD: D*

In a proportion whose terms are homogeneous, the sum of the greatest and least exceeds the sum of the other two. Let the proportion be A: B :: C: D, and suppose, first, that an antecedent, as A, is the greatest term; then (prop. vii. and ix.) D will be the least, and it is required to prove that (AD) > (B+ C).

-

By the preceding proposition A: A-B::C: C – D, therefore (prop. ix.) since A > C, (A — B) > (C — D). To each of these unequals add B + D, and there results (A + D) > (B+C).

* In inferring this proportion from the original, the older Geometers say "by division," and the next proportion following, or rather that taken inversely, is said to be inferred "by conversion."

Hence,

Next, let a consequent be the greatest as B; then since by inversion B: A:: D: C, C must be the least. reasoning as before (B + C) > (A + D).

COR. In the proportion A: B:: B: C, (A + C) > 2B; so that the MEAN TERM of three proportionals is less than half the sum of the EXTREMES.

PROP. XVI. THEOR.

If there be Two SERIES of magnitudes, such that the first term is to the second in the first series, as the first to the second in the other series; and the second term to the third in the former as the second to the third in the latter, and so on; then as the first term is to the last in the one series so is the first to the last in the other.

First, let there be three magnitudes in each series, viz. A, B, C and D, E, F furnishing the proportions A: B:: D: E and B C E : F* then

A CD: F.

From the first proportion (props. viii., and vi. cor.)

from the second B nC E: nF,

the multiples being any whatever.

Let mA > nC, then a multiple sufficiently great of mA can be found that will contain B oftener than the like multiple of nC. Let pmA be such a multiple; then pmA contains B oftener than pnC contains B.

Austin, in his "Examination of the first six books of Euclid's Elements," applies alternation to these proportions in order to improve Euclid's demonstration, and thus, instead of Euclid's proposition, he demonstrates, unconsciously, only a very particular case, viz. that in which all the quantities in both series are of the same kind.