Again, the test is also satisfied if the multiples of any multiple (in) of A are distributed among the multiples of any multiple (n) of B in the same manner as the multiples of In o among the multiples of n n : for instance, if the multiples of 3 A be distributed among those of 5 B in the same manner as the multiples of 3 c among those of 5 D. Let 11 A and 11 o be given for examination; take any multiple of 3 greater than 5, say 3 x 3, or 9, and examine 11 (9 e) and 11 (9 c), or 33 (3 A) and 33 (3 c). Let 33 (3 e) lie between 27 (5B) and 23 (5 is) ; then by hypothesis 33 (3 c) lies between 27 (5D) and 28 (5 D). Divide all by 9, and we find that 11 A lies' between l 5 B and 153n, while 11 o bee between 15 i and 153 D. Hence 11 A lies between 15B and 16n, while 11 o lies in the same interval among the multiples of D ; and in the same manner any other instance may be proved.
The principles of the fifth book of Euclid are by many supposed to be inevitably connected with the apparatus of straight lines drawn parallel to one another, by which Euclid represents his magnitudes and their multiples. This is not the case; and the simple expression of magnitudes by Large letters, and of numerical multipliers by small ones, will very much facilitate the demonstrations, without altering anything but mere modes of expression.
The next point to be considered ie the infinite character of the definition of proportion ; four magnitudes are not to be called pro portional, until it is shown that every multiple of A falls in the same interval among tho multiples of B, in which the same multiple of c is found among the multiples of D. So that this definition is n negative one, like that of parallel lines, which may be thus stated : two lines are parallel when every point of one of them, however far produced, is on one side of the other. We might expect then to find that the teat of disproportion is simple and positive, and an examination of the illustration already produced will confirm this.
Suppose that the distribution of the railings among the columns should be found to agree in the model and the original as far as the millionth railing. This proves, as we have seen, only that the railing distance of the model does not err by the millionth part of the corre sponding column-distance; for if it did err so much, the multiplication of the error a million-fold would have placed the millionth railing (if none before it) wrong by at least one interval. It is then obvious that no examination of individual cases, however extensive, will enable an observer of the construction and its model to affirm the proportion or deny disproportion : all that it can do is to enable him to fix limits (which he may make as small as he pleases) to the disproportion, if any. But a
single natant* may enable him to deny proportion or affirm dispropor tion, and also to state which' way the disproportion lies.. Let the 19th railing in the original fall beyond the Ilth column, while the 19th railing of the (so-called) model does not come up to the 11th column; It fol lows from this one instance, that the railing-distance of the model is too small relatively to the column-distance, that the column-distance is too great relatively to the railing-distance ; that is, the proportion of r to cis less than that of a to o, or the proportion of c to r is greater than that of a to tt. Similarly, with respect to two straight lines, no examination of pairs of points, one in each, will enable the examiner to affirm their parallelism or deny their intersection ; while, at the some time, tha examination of one pair of points may enable him to affirm Intersection or deny parallelism. Hence it appears that, obvious as the notion of proportion may be, it is more easy, in a mathematical long and intricate propositions, we should agree with those who would refuse to place those propositions before beginners, on the ground that unimpeachable demonstration is already given, namely, an assumption which those who would rather dispense with it do not deny to be both true and easy (or capable of being made easy), and logic which is un contested and incontestable. But the vice of the system which is sub stituted for that of Euclid, consists in the entrance of an assumption which is not true ; for to reason upon all magnitudes as if they were commensurable, and to assert conclusions derived from such reasonings, is to assume that all magnitudes are commensurable,which is not true. The method of Lacroix, as above explained, is sound as far as it goes : he asserts that the propositions of geometry are sensibly true, or if false, imperceptibly false, and this he proves. But he makes geometry cease to be an exact science. That of Legendre, on the other hand, though it proves no more than that of Lacroix, professes to prove more : it treats geometry as an exact science, while it avowedly states that an assumption may be made which is demonstrably incorrect.