21. If there be four proportionals, the extremes multiplied into one another, and the means multiplied into one another, will make equal products. Let A. B :: C. D be proportionals; I say A D is equal to B C. For the proportion of A D to B C is compounded, by [art. 13], of the proportions of A to B, and D to C, that is, its inverse B to A; and therefore, by [art. 14], this compounded proportion is the proportion of equality; and therefore also, the proportion of A D to B C is the proportion of equality. Wherefore they are equal.
22. If there be four quantities, and the proportion of the first to the second be duplicate of the proportion of the third to the fourth, the product of the extremes to the product of the means, will be as the third to the fourth. Let the four quantities be A, B, C and D; and let the proportion of A to B be duplicate of the proportion of C to D, I say A D, that is, the product of A into D is to B C, that is, to the product of the means, as C to D. For seeing the proportion of A to B is duplicate of the proportion of C to D, if it be as C to D, so D to another, E, then A. B :: C. E will be proportionals; for the proportion of A to B is by supposition duplicate of the proportion of C to D; and C to E duplicate also of that of C to D by the definition, [art. 15]. Wherefore, by the last [article], A E or A into E is equal to B C or B into C; but, by coroll. IV. [art. 6], A D is to A E as D to E, that is, as C to D; and therefore A D is to B C, which as I have shown is equal to A E, as C to D; which was to be proved.
Moreover, if the proportion of the first A to the second B be triplicate of the proportion of the third C to the fourth D, the product of the extremes to the product of the means will be duplicate of the proportion of the third to the fourth. For if it be as C to D so D to E, and again, as D to E so E to another, F, then the proportion of C to F will be triplicate of the proportion of C to D; and consequently, A. B :: C. F will be proportionals, and A F equal to B C. But as A D to A F, so is D to F; and therefore, also, as A D to B C, so D to F, that is, so C to E; but the proportion of C to E is duplicate of the proportion of C to D; wherefore, also, the proportion of A D to B C is duplicate of that of C to D, as was propounded.
23. If there be four proportionals, and a mean be interposed betwixt the first and second, and another betwixt the third and fourth, the first of these means will be to the second, as the first of the proportionals is to the third, or as the second of them is to the fourth. For let A. B :: C. D be proportionals, and let E be a mean betwixt A and B, and F a mean betwixt C and D; I say A. C :: E. F are proportionals. For the proportion of A to E is subduplicate of the proportion of A to B, or of C to D. Also, the proportion of C to F is subduplicate of that of C to D; and therefore A. E :: C. F are proportionals; and by permutation A. C :: E. F are also proportionals; which was to be proved.
24. Any thing is said to be divided into extreme and mean proportion, when the whole and the parts are in continual proportion. As for example, when A + B. A. B are continual proportionals; or when the straight line A C is so divided in B, that A C. A B. B C are in continual proportion. And if the same line A C be again divided |A B C
——+——+——
D| in D, so as that A C. C D. A D be continual proportionals; then also A C. A B. A D will be continual proportionals; and in like manner, though in contrary order, C A. C D. C B will be continual proportionals; which cannot happen in any line otherwise divided.
25. If there be three continual proportionals, and again, three other continual proportions, which have the same middle term, their extremes will be in reciprocal proportion. For let A. B. C and D. B. E be continual proportionals, I say A. D :: E. C shall be proportionals. For the proportion of A to D is compounded of the proportions of A to B, and of B to D; and the proportion of E to C is compounded of those of E to B, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by equality, A. D :: E. C are proportionals.
Comparison of arithmetical and geometrical proportion.
26. If any two unequal quantities be made extremes, and there be interposed betwixt them any number of means in geometrical proportion, and the same number of means in arithmetical proportion, the several means in geometrical proportion will be less than the several means in arithmetical proportion. For betwixt A the lesser, and E the greater extreme, let there be interposed three means, B, C, D, in geometrical proportion, and as many more, F, G, H, in arithmetical proportion; I say B will be less than F, C than G, and D than H. For first, the difference between A and F is the same with that between F and G, and with that between G and H, by the definition of arithmetical proportion; and therefore, the difference of the proportionals which stand next to one another, to the difference of the extremes, is, when there is but one mean, half their difference; when two, a third part of it; when three, a quarter, &c.; so that in this example it is a quarter. But the difference between D and E, by art. [17], is more than a | A A
- -
B F
— —
C G
–— –—
D H
—— ——
E E
——— ———| quarter of the difference between the extremes, because the proportion is geometrical, and therefore the difference between A and D is less than three quarters of the same difference of the extremes. In like manner, if the difference between A and D be understood to be divided into three equal parts, it may be proved, that the difference between A and C is less than two quarters of the difference of the extremes A and E. And lastly, if the difference between A and C be divided into two equal parts, that the difference between A and B is less than a quarter of the difference of the extremes A and E.
From the consideration hereof, it is manifest, that B, that is A together with something else which is less than a fourth part of the difference of the extremes A and E, is less than F, that is, than the same A with something else which is equal to the said fourth part. Also, that C, that is A with something else which is less than two fourth parts of the said difference, is less than G, that is, than A together with the said two-fourths. And lastly, that D, which exceeds A by less than three-fourths of the said difference, is less than H, which exceeds the same A by three entire fourths of the said difference. And in the same manner it would be if there were four means, saving that instead of fourths of the difference of the extremes we are to take fifth parts; and so on.
27. Lemma. If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. Let B be added to and substracted from the quantity A; so that A - B be the remainder, and A + B the aggregate; and again, let C, a greater quantity than B, be added to and substracted from the same A, so that A - C be the remainder and A + C the aggregate; I say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A :: A - C. A is an hyperlogism of a greater antecedent to the same consequent; and therefore A - B. A + B :: A - C. A + C is a much greater hyperlogism, being made of a greater antecedent to a less consequent.