Elements of algebra Volume 1 - Softcover

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1810 Excerpt: ...In the same manner, as 12--7=9--4, so also 12--9=7--4. 393. We may in every arithmetical proportion put the second term also in the place of the first, if we make the same transposition of the third and fourth; that is, if a--b=p--q, we have also b--a =q--'p; for b--a is the negative of a--b, and q--p is also the negative of p--q; and thus, since 12--7 =2--4, we have also, 7--12=4--9. 394. But the most interesting property of every arithmetical proportion is this, that the sum of the second and third term is always equal to the sum of the first and fourth. This property, which we must particularly consider, is expressed also by saying that the sum of the means is equal to the sum of the extremes; thus, since 12--7=9--4, we have 7+9 = 12+4; thersum being in both eases 16. 395. In order to demonstrate this principal property, let a--b-=p--q; then if we add to both b+q, we have a+q = b+p; that is, the sum of the first and fourth terms is equal to the sum of the second and third: and inversely, if four numbers, a, b, p, q, are such that the sum of the second and third is equal to the su.m of the first and fourth, that is, if b--p =a+q, we conclude, without a possibility of mistake, that those numbers are in arithmetical.proportion, and that d--b=p--q; for, since a+q=b--p, if we subtract from both sides b+q, we 'obtain a--b =p-q. Thus the numbers 18, 13, 15, 10, being such that the sum of the means (13+ 15=28) is equal to the sum of the extremes (18+10 = 28), it is certain that they also form an arithmetical proportion; and consequently, that 18--13=15--10. 396. It is easy, by means of this property, to resolve the following question. The three first terms of an arithmetical proportion being given, to find the fourfh? Let a, b, p, be the three first terms, and l...