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Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB each equal to C, as there are in DE each equal to F: let AB be divided into magnitudes, each equal to C, viz., AG, GH, HB; and DE into magnitudes, each equal to F, viz., DK, KL, LE. Then the number of the first AG, GH, HB, shall be equal to the number of the last DK, KL, LE. And because AG, GH, HB are all equal, and that DK, KL, LE are also equal to one another; therefore (V. 7.) AG:DK::GH:KL, and as HB : LE. And (V. 12.) as one of the antecedents to its consequent, so are all the antecedents together to all the consequents together ; wherefore as AG:DK:: AB:DE: but AG is equal to C, and DK to F. Therefore, as C:F:: AB: DE: wherefore magnitudes, &c.

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PROP. XVI. THEOR.

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If four magnitudes of the same kind be proportionals, they are also proportionals when taken alternately.

Let A:B::C: D: then, alternately, A:C::B: D.

Take of A and B any like multiples whatever E and F; and of C and D take any whatever G and H: and because E is the same multiple of A, that F is of B, and that magnitudes (V. 15.) have the same ratio to one another which their like multiples have; therefore A:B::E:F: but as A:B:: C:D: wherefore (V.11.) as C:D::E: F. Again, (V. 15.) because Ġ, H are like multiples of C, D, as C:D::G:H; but as C:D::E:F: wherefore (V. 11.) as E : F::G:H. But, (V. 14.) when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth; if equal, equal; and if less, less : therefore, if E be greater than G, F likewise is greater than H; if equal, equal; and if less, less: and E, F are any like multiples whatever of A, B; and G, H any whatever of C, D. Therefore (V. def. 5.) A:C::B:D; wherefore, if four magnitudes, &c.

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PROP. XVII. THEOR. If four magnitudes be proportionals; then, by division, the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth.

Let AB: BE::CD:DF; then AE: EB :: CF:FD.

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Take of AE, EB, CF, FD any like multiples whatever GH, HK, LM, MN; and, again, of EB, FD, take any like multiples whatever KX, NP. Then, because GH is the same multiple of AE, that HK is of EB; therefore (V. 1.) GH is the same multiple of AE, that GK is of AB : but GH is the same multiple of A E, that LM is of CF; wherefore GK is the same multiple of AB, that LM is of CF. Again, because LM is the same multiple of CF, that MN is of FD ; therefore (V. 1.) LM is the same multiple of CF, that LN is of CD. But LM was shown to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, LN are like multiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; ảnd that KX is also the same multiple of EB, that NP is of FD; therefore (V. 2.) HX is the same multiple of EB, that MP is of FD. And, because AB: BE:: CD:DF, and that of AB and CD, GK and LN are like multiples, and of EB and FD, HX and MP are like multiples ; if GK be greater than HX, then (V. def. 5.) LN is greater than MP; if equal, equal ; and if less, less. But if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, that if GH be equal to KX, LM is likewise equal to NP; and if less, less : and GH, LM are any like mul. tiples whatever of AE, CF, and KX, NP are any whatever of EB, FD. Therefore (V. def. 5.) as AE: EB::CF: FD. Wherefore, if four magnitudes, &c.

PROP. XVIII. THEOR.

If four magnitudes be proportionals; then, by composition, the first and second together are to the second, as the third and fourth together are to the fourth.

Let AE: EB:: CF:FD; then, AB: BE::CD:DF.

Take of AB, BE, CD, DF any like multiples GH, HK, LM, MN; and again, of BE, DF, take any like multiples KO, NP. Then, because KO, NP are like multiples of BE, DF; and that KH, NM are also like multiples of BE, DF, if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, NP, likewise the multiple of DF, shall be greater than MN, the multiple of the same DF; and if Ko be equal to KH, NP shall be equal to NM; and if less, less.

First, let KO not be greater than KH, therefore NP is not

greater than NM: and because GH, HK are like multiples of AB, BE, and that AB is greater than BE, therefore (V. ax. 3.) GH is greater than HK; but KO is not greater than KH, wherefore GH is greater than Ko. In like manner it may be shown, that LM is greater than NP. Therefore, if KO be not

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greater than KH, then GH, the multiple of AB, is always greater than KO, the multiple of BE; and likewise LM, the multiple of CD, greater than NP, the multiple of DF.

Next, let KO be greater than KH: therefore, as has been shown, NP is greater than NM. And because the whole GH is the same multiple of the whole AB, that HK is of BE, the remainder GK is the same multiple (V.5.) of the remainder AE, that GH is of AB: which is the same that LM is of CD. In like manner, because LM is the same multiple of CD, that MN is of DF, the remainder LN is the same multiple (V. 5.) of the remainder CF, that the whole LM is of the whole CD. But it was shown that LM is the same multiple of CD, that GK is of AE; therefore GK is the same multiple of A E, that LN is of CF; that is, GK, LN are like multiples of AE, CF. And because KO, NP are like multiples of BE, DF, if from KO, NP there be taken KH, NM, which are also like multiples of BE, DF, the remainders HO, MP are (V. 6.) either equal to BE, DF, or like multiples of them. First, let HO, MP be equal to BE, DF; and because AE: EB :: CF: FD, and that GK, LN are like multiples of AE, CF; then (V. 4. cor.) GK: EB :: LN : FD : but HO is equal to EB, and MP to FD; wherefore GK:HO:: LN: MP. Therefore (V. E.) if GK be greater than HO, LN is greater than MP; if equal, equal ; and if less, less.

But let HO, MP be like multiples of EB, FD: and because AE: EB :: CF:FD, and that of AE, CF are taken like multiples GK, LN ; and of EB, FD, like multiples HO, MP; therefore (V. def. 5.) if GK be greater than HO, LN is greater than MP; if equal, equal; and if less, less; which was likewise shown in the preceding case. If therefore GH be greater than KO, taking KH from both, GK is greater than HO; wherefore also LN is greater than MP; and consequently adding NM to both, LM is greater than NP: therefore, if GH be greater than KO, LM is greater than NP. In like manner it may be shown that if GH

be equal to KO, LM is equal to NP; and if less, less. And in the case in which KO is not greater than KH, it has been shown that GH is always greater than KO, and likewise LM than NP: but GH, LM are any like multiples of AB, CD, and KO, NP are any whatever of BE, DF; therefore (V. def. 5.) as AB:BE::CD: DF. If, therefore, four magnitudes, &c.

PROP. XIX. THEOR.

If the first be to the second, as a part of the first to a part of the second ; the remaining part of the first is to the remaining part of the second, as the first to the second.

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Let AB:CD:: AE: CF; then EB: FD :: AB: CD.

Because AB : CD :: AE : CF; alternately, (V. 16.) BA ; AE:: DC : CF: and therefore, (V. 17.) by division, as BE:EA :: DF:FC; and alternately, as BE:DF:: EA: FC. But (hyp.) as AE:CF:: AB: CD; therefore (V. 11.) also BE: DF :: AB : CD: wherefore, if the first, &c.

Cor. If the first be to the second, as a part of the first is to a part of the second; the remaining part of the first is to the remaining part of the second, as the other part of the first to the remaining part of the second. The demonstration is contained in the preceding.

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PROP. E. THEOR.

If four magnitudes be proportionals, they are also proportionals by conversion; that is, the first is to its excess above the second, as the third to its excess above the fourth.

Let AB : BE::CD: DF; then BA: AE::DC: CF. (See the figure to the last proposition.)

Because AB: BE:: CD: DF, by division (V. 17.) AE:EB:: CF : FD; and (V. B.) by inversion, BE : EA : : DF: FC. Wherefore, (V. 18.) by composition, BA : AE :: DC: CF. If, therefore, four magnitudes, &c.

Cor. It would be proved in a similar manner, that the first is to the sum of the first and second, as the third to the sum of the third and fourth.

PROP. XX. THEOR.

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If there be three magnitudes, and other three, which, taken two and two, have the same ratio; if the first be greater than the third, the fourth is greater than the sixth ; if equal, equal ; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three, which, taken two and two, have the same ratio, viz., A:B::D:E; and B:C::E:F: if A be greater than C, D is greater than F; if equal, equal; and if less, less.

Because A is greater than C, and B is any other magnitude, and that (V. 8.) the greater has to the same magnitude a greater ratio than the less has ; therefore A has to B a greater ratio than C has to B. But as D:E::A:B; therefore (V. 13.) D has to E a greater D Е Р ratio than C to B; and because B:C::E: F, by inversion, C:B::F: E; and D was shown to have to E a greater ratio than C to B; therefore (V. 13. cor.) D has to E a greater ratio than F to E: but (V. 10.) the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two: D is therefore greater than F.

Secondly, let A be equal to C; D is also equal to F. Because A and C are equal to one another, (V. 7.) A: B::C:B: but A:B::D:E; and C:B::F: E; wherefore also (V. 11.) D:E::F:E; and therefore (V.9.) D is equal to F.

Next, let A be less than C; D is also less than F. For C is greater than A, and, as was shown in the first case, C:B::F:E, A and in like manner B:A::E:D; there

E F fore F is greater than D, by the first case ; that is, D is less than F. Therefore, if there be three magnitudes, &c.

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PROP. XXI. THEOR.

If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first be greater than the third, the fourth is greater than the sixth ; if equal, equal ; and if less, less.

Let A, B, C be three magnitudes, and D, E, F other three,

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