require it), then if we have the equations C G it will follow that we must have D H' A E B = B FC For, since' A, B, C are one kind of quantities, we can .divide A and B by C, and we shall get It is evident, from what has been done, that the same reasoning and conclusion will hold when there are five quantities of each of the above kinds, and when there are six of each kind, and so on, to any (the same) number of each kind. Since the equations used are equivalent to proportions, we say, if we have two ranks of quantities, such that the quantities in each rank are of one kind, and the number of quantities in one rank is the same as the number of quantities in the other rank; and if the first quantity in the first rank has the same ratio to the second that the first quantity of the second rank has to the second (of that rank), and if the second quantity in the first rank has the same ratio to the third that the second quantity of the second rank has to the third (of that rank), and so on in order for all the quantities of the two ranks; then we infer from what has been done (which is called "ex æquali"), that the first quantity of the first rank has the same ratio to the last of that rank, as the first of the second rank has to the last of that rank. (16.) Let A, B, C denote three quantities of one kind, and D, E, F three quantities of one kind (which may also be of the same kind as A, B, C, if required), such that we have A E B D then we shall also have the = = B FC E' the equations For if we represent any two positive integers by m and n, we evidently get from the supposed equations MA nE = and or for this equation (by inversion of proportion, as has been shown in (12), we may write nC nE = mB mD If we now suppose that mA is greater than nC, we shall of course have mA nE and nC nE mB mD' nE mD' greater than consequently mD must be greater than nF; so that when mA is greater than nC, mD is greater than F. In like manner, if mA equals nC, we have mD equal to nF; and if mA is less than nC, we have mD less than nF. Hence we have the four multiples mA, nC, mD, nF such that when the first is greater, equal to, or less than the second, the third is in like manner greater, equal to, or less than the fourth; consequently, from (14) we have the proportion A: C::D: F, or, which is the same, we A D have the equation C F Again, let A, B, C, D be four quantities of one kind, and E, F, G, H four quantities of one kind, such that we have A GBF C E B H'C G'D=F' For, by what has just been proved, the first two of the last of the supposed equations, give in the same way the It is easy to see that if we have two ranks of quantities, such that there are as many quantities in one rank as in the other, and that the quantities in each rank are all of one kind; then if conditions or equations like to those supposed in the cases of three and four quantities in a rank have place, it will follow that the first of the first rank divided by the last of the first rank, will give the same quotient as the first of the second rank divided by the last of the second rank. For when there are five quantities in each rank, they can be reduced as in the case of four quantities in a rank to the case of three quantities in a rank; and in the case of six quantities in a rank the same reductions will have place, and so on, to any extent. Hence, if we have the same number of quantities in two ranks; such that the first quantity in the first rank is to the second quantity in that rank as the last quantity but one in the second rank is to the last quantity in that rank; and the second quantity in the first rank is to the third quantity in that rank, as the last quantity but two in the second rank is to the last quantity but one in that rank, and so on, for all the quantities of the two ranks, taking them in a cross order; then we shall have the first quantity of the first rank is to the last quantity of that rank as the first quantity of the second rank is to the last quantity of that rank; as is evident, since the proportions stated are virtually the same as the equations which we have used in the proof and conclusion of our proposition. Remark. The proposition here proved is usually quoted by the words "ex æquali in proportione perturbata;" or "ex equo perturbato." See Simson's Euclid, Book V., Prop. XXIII. (17.) If the antecedents of one proportion are the same as the consequents or antecedents of any other proportion, then shall the remaining terms in the proportions constitute a proportion; such that the two terms of one of the proportions shall be the antecedents and the two remaining terms of the other proportion shall be the consequents. For let the proportions be A: B:: C:D and B: E:: D: F; then, placing them in two ranks, we have A, B, E for one rank, and C, D, F for the other rank. Hence, by (15) (which is usually cited by the words "ex equali"), since the quantities in the two ranks are proportional when taken two by two in order, we get the proportion A: E:: C: F, as required. In the proof, the equal terms are the consequents in one of the proportions, and the antecedents in the other; if, however, they should be the consequents or antecedents in both proportions, then, by (12), we can invert the terms of one of the proportions, and then the proof of the proposition will be the same as before. (18.) If the extremes of one proportion are the same with the means of another proportion, or if the extremes or means of two proportions are the same, then shall the remaining terms of the two proportions constitute a proportion whose means shall be the remaining terms of one of the propor tions, and extremes the remaining terms of the other proportion. For let A: B:: E: F, B: C:: D: E be the two proportions; then, writing them down in two ranks, we have A, B, C for one rank, and D, E, F for the other rank of quantities. Hence, since the quantities in the two ranks are proportional when taken two by two in each rank in a cross order, we have by (16) (or "ex equo perturbato"), A: C:: D: F, as was to be shown. If the extremes or means are the same in the given proportions, then, by (12), invert the terms of one of them, and proceed as before. B is contained once in B; and, in the same way, to be proved. In other terms, if we have the proportion A: B: C: D, we shall also have the proportion A + B : B :: C+D: D. In like manner, by inverting the terms of the proportion A: B:: C:D by (12), we shall have the pro C + D = portion B: A: D: C; hence (as before) we shall have the proportion A+B: A:: C+ D: C. Hence, we say (by composition) that the sum of the first two terms of any proportion is to either of the two terms as the sum of the last two terms is to that term of the last two terms that corresponds to the term that is taken for the second term of the proportion. (20.) If we have the equation A C or the proportion = B A B C D, then, if A is greater than B, we shall also have the equation A B C-D A-B: B:: C-D: D. For or the proportion D A B B B D portion AB: B:: C-D: D. From the proportion A-B: B:: C-D: D, we get by (19) A- B+B: AB:: C-D+D: CD, or (since B+B=0, and - D +D0) we have A: A- B:: C: C-D, or by (12) A B: A: C-D: C. If we have B greater than A, we may - A from (12); hence, as before, we shall have B-A: A :: D - C: C, or B A: B: DC: D. Hence (by division), we say that the difference of the first two terms of any proportion is to either of the first two terms as the difference of the last two terms is to that term of the two which corresponds to the term that is taken for the second term of the proportion. (21.) From the equation when A is greater than B; or, which comes to the same, from the proportion A: B:: C : D we shall have the proportion A + B : A − B :: C + D : C − D, when A is greater than B. For from (19) we get the propor |