that is, A: C: A: B2, or A is to C in the duplicate ratio of A to B. Again, if A, B, C, D, be in continued proportion, or A is to D in the triplicate ratio of A to B, and so on with any number of quantities in continued proportion. 13. In proportionals, the antecedent terms are called homologous to one another, as also the antecedents to one another. Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals. 14. Permutando, or alternando, by permutation, or alternately; this word is used when there are four proportionals, and it is inferred, that the first has the same ratio to the third, which the second has to the fourth; or that the first is to the third, as the second to the fourth: as is shown in the 16th proposition of this book. 15. Invertendo, by inversion; when there are four proportionals, and it is inferred, that the second is to the first, as the fourth to the third. Proposition B, book 5. 16. Componendo, by composition; when there are four proportionals, and it is inferred, that the first, together with the second, is to the second, as the third together with the fourth, is to the fourth. Proposition XVIII., book 5. 17. Dividendo, by division; when there are four proportionals, and it is inferred, that 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. Proposition XVII., book 5. 18. Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Proposition E, book 5. 19. Ex æquali (sc. distantiâ), or ex æquo, from equality of distance; when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the first is to the last of the others: "Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken two and two." 20. Ex æquali, from equality; this term is used simply by itself, when the first magnitude is to the second of the first rank, as the first to the second of the other rank; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order, and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in Proposition XXII., book 5. 21. Ex æquali, in proportione; perturbata, seu inordinata, from equality, in perturbate or disorderly proportion (Prop. 4, Lib. II. Archimedis de sphæra et cylindro); this term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank: and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank: and so on in a cross order: and the inference is as in the 19th definition. It is demonstrated in Proposition XXIII., book 5. The following table will serve to illustrate and explain the foregoing seven last definitions. If any four magnitudes be in proportion, so that A: B:: C Then Permutando or Alternando A: C:: : D B : D The terms subduplicate, subtriplicate, and sesquiplicate ratios being frequently employed in astronomy should be defined. If three quantities be in continued proportion, the first is said to have to the second the subduplicate ratio of that which the first has to the third. Thus, if A, B, C, are in continued proportion, A is said to have to B the subduplicate ratio of that which A has to C, and may be expressed algebraically A: B:: A c1. : If four quantities be in continued proportion, the first is said to have to the second the subtriplicate ratio of that which the first has to the fourth. Thus, if A, B, C, D, are in continued proportion, A is said to have to B the subtriplicate ratio of that which A has to D, and may be expressed algebraically, A: B:: A3 : D3. A sesquiplicate ratio is the ratio compounded of the simple ratio and the subduplicate, and may be expressed algebraically, A: B :: A cl. : AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. Or if equals be multiplied by the same, the products are equal. 2. Those magnitudes of which the same, or equal magnitudes,` are equimultiples, are equal to one another. Or if equals be divided by the same, the quotients are equal. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. Of two magnitudes that one of which a multiple is greater than the same multiple of the other, is the greater. In the following propositions lines are employed by Euclid to represent proportional magnitudes, but it should be understood that any similar magnitudes might have been employed, such as plane figures, solid bodies, or angles. ROPOSITION I. THEOREM.-If any number of magnitudes be equimultiples of as many others, each of each; what multiple soever any one of the first is of its part, the same multiple shall all the first magnitudes taken together be of all the others taken together. Let any number of magnitudes AB, CD, be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the same multiple shall AB and CD together be of E and F together. DEMONSTRATION. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD, equal each of them to F; the number, therefore, of the magnitudes CH, HD is equal to the number of the others AG, GB (a). And because AG is equal to E, and CH to F, therefore AG and CH together are equal to E and F together (b). For the same reason, GB and HD together are equal to E and F together; wherefore, as many magnitudes as are in AB equal to E, so many are there in AB and CD together equal to E and F together. Therefore, whatever multiple AB is of E, the same multiple are AB and CD together, of E and F together; and the same demonstration would hold if the number of magnitudes were greater than two. Therefore, if any number of magnitudes, &c. A G B H D E FI (a) Hypoth. (b) I. Ax. 2. SCHOLIUM. In order to the elucidation of Euclid's demonstrations we shall append to each proposition an algebraical investigation and proof, preserving his train of reasoning unaltered. THEOREM. If A, B, C, &c., be equimultiples of a, b, c, fc., then whatsoever multiple A is of a, the same multiple is A+B+C+ &c., of a+b+c+ &c. Let A contain n parts each equal to a, then A = na and because B, C, &c., are the same multiples of b, c, &c., that A is of a, therefore therefore A + B + C + &c. = n a + n b + nc+ &c. = = a + a + a + &c. to n terms (a + b + c + &c.) + = n (a + b + c + &c.) + b + b + b + &c. to n terms PROPOSITION II. THEOREM.-If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first, together with the fifth, be the same multiple of the second, that the third, together with the sixth, is of the fourth. Let AB the first be the same multiple of C the second, that DE the third is of F the fourth, and BG the fifth the same multiple of C the second, that EH the sixth is of F the fourth; then shall AG the first together with the fifth, be the same multiple of C the second, that DH the third together with the sixth, is of F the fourth. A D E DEMONSTRATION. Because AB is the same multiple of C that DE is of F, there are as many magnitudes in AB equal to C as there are in DE equal to F; in like manner, as many as there are in BG equal to C, so many are there in EH equal to F; therefore, as many as there are in the whole AG equal to C, so many are there in the whole DH equal to F; therefore, AG is the same multiple of C that DH is of F, that is, AG, the first and fifth together, is the same multiple of the second C, that DH, the third and sixth together, is of the fourth E. Therefore, if the first magnitude, &c. B G C H |