# The Doctrine of Proportion Clearly Developed: On a Comprehensive, Original, and Very Easy System; Or, The Fifth Book of Euclid Simplified

J. Williams, 1841 - Euclid's Elements - 98 pages

### Contents

 Section 1 1 Section 2 23 Section 3 45 Section 4 55 Section 5 56 Section 6 58 Section 7 68
 Section 8 78 Section 9 82 Section 10 85 Section 11 92 Section 12 98 Section 13 Copyright

### Popular passages

Page 10 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any...
Page 2 - Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part, or parts, one quantity is of the other.
Page 58 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is usually cited by the words
Page 62 - If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio ; the first shall have to the last of the first magnitudes, the same ratio which the first of the others has to the last. NB This is usually cited by the words "ex sequali,
Page 18 - IF the first be the same multiple of the second, or the same part of it, that the third is of the fourth ; the first is to the second, as the third is to the fourth...
Page 32 - THAT magnitude which has a greater ratio than another has to the same magnitude, is the greater of the two : and that magnitude, to which the same has a greater ratio than it has to another magnitude, is the less of the two.
Page 21 - IF the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth...
Page 55 - 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 shall be greater than the sixth ; and if equal, equal ; and if less, less...
Page 14 - IF one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other ; the remainder shall be the same multiple of the remainder, that the whole is of the whole.
Page 73 - L : and the same thing is to be understood when it is more briefly expressed, by saying A has to D the ratio compounded of the ratios of E to F, G to H, and K to L. In like manner, the same things being supposed, if M has to N the same ratio which A has to D ; then, for shortness...