a circle, we do, in fact, make a statement, which, brief as it is, is comprehensive of all the properties of the circle. It is the business of science to develop these properties. Without a process of mathematical reasoning this development could not be effected; and without first admitting the axioms, or certain indemonstrable though obvious principles, this reasoning could not proceed. The circle, as presented by its definition, is the raw material upon which we operate, and mathematical reasoning is the machinery we employ to evolve from it all that it contains. And, as elements of this machinery, the axioms as much enter as do the simple mechanical powers into every piece of physical mechanism. There is no doubt, however, that propositions are sometimes called axioms, which really admit of demonstration, or which may be shown to be reducible to some other, really an ultimate principle. Such is that called axiom xi., which is reducible to axiom ix. See "Young's Elements of Geometry," page 7. Note (B), page 38. Proof of Euclid's 12th Axiom, by M. Bertrand. The reasoning of M. Bertrand, to show that two straight lines which make with a third two interior angles, that are together less than two right angles, may be produced till they meet, is as follows: Let the straight lines CA, DB make with AB interior angles at A and B, whose sum is less than two right angles; then AC, BD may be produced till they meet. Let the angle EAB make with the angle DBA two right angles; then AC must lie within the sides of the angle EAB. Take the angle E'A'B' equal to EAB, and E'A'C' equal to EAC. Then it is plain that, whatever be the magnitude of the angle E'A'C', a multiple of it may be taken so great as to exceed the angle E'A'B'; in other words, the angle E'A'C' will, by some finite number of repetitions about the vertex A', at length fill up the angle E'A'B'; and, consequently, the unlimited space comprised between the sides A'E', A'C' indefinitely prolonged, will, by such repetition, entirely fill up the unlimited space comprised between the lines A'E', A'B' indefinitely prolonged. Therefore, the space EAC will, by repetition, fill up the space EАВ. Let us now consider the unlimited space or band EABD, which we may repeat as often as we please, upon the prolongation of the base AB; for if BF be taken equal to AB, and the angle BFG be made equal to ABD, it is obvious that the band EABD will, upon application, entirely coincide with the band DBFG; for the angles at their bases are equal each to each, and the bases themselves are equal. We may thus, therefore, multiply these bands indefinitely; but we shall never be able to fill up the unlimited space comprised between the lines AE, AB. It has been shown, however, that the space between AE, AC, by being repeated a limited number of times, will fill up the unlimited space between AE, AB. It follows, therefore, that the space between AE, AC must exceed the space EABD, and, therefore, cannot possibly be included in that space, which must, however, be the case if AC do not meet BD; AC, therefore, must necessarily meet BD. The above demonstration, though long known on the Continent, was, we believe, first introduced to the English student by the Editor of this work, in his "Elements of Geometry," 1827, to which the student is referred for a detailed account of the different methods which have been proposed for removing the difficulty in the theory of parallel lines. The following plausible though inconclusive demonstration of the 12th axiom occurred to the Editor, some years ago; and he is induced to insert it here, as rather an instructive example of specious reasoning. Let AB be parallel to CD. Through any point P in AB no other parallel to CD can be drawn. Take any point E in CD, and draw PE. If between the sides of the angle EPB a line from P parallel to CD could be drawn, then, since all such, as PQ, PR, &c. cannot be parallel, there must be some one, as PR, which will be the last of the meeting lines. But CR may be prolonged; Z in the prolongation. Then, since P, Z we have a meeting line PZ beyond PR. hypothesis, the last of the meeting lines; hence that hypothesis is absurd. take, therefore, may be joined, But PR is, by In like manner may it be shown that no other line within the angle EPB is a final, or non-meeting line; hence the only non-meeting line is AB. We leave the student to detect the fallacy of this reasoning; merely remarking that it is of a similar kind to that unconsciously committed by Legendre, in proposition ix. of the Fourth Book of his Geometry, and which, though long ago pointed out by the Editor of this work, continues to be copied by writers on Geometry. Note (C), page 85. The error in Euclid's demonstration of this proposition (if, indeed, it be his demonstration,) arises principally from his making his reasoning applicable only to that particular position of the fictitious centre assumed in the diagram; and which is such as to cause a line through it, from the proposed point, to leave all three of the lines, hypothetically equal, upon one side of it: and the conclusion is made to depend on the circumstance of the third line thus being the most remote from that through the fictitious centre. It is plain, however, that the fictitious centre may be so assumed that only two of the equal lines may be on one side of the line from the proposed point through it. And two being necessarily on one side of the line through the supposed centre, and being also equal, establishes the inference of. Euclid by the proposition he has referred to, viz. the 7th of the 3d Book. The demonstration of Euclid will, therefore, be rectified by omitting, in his diagram, the third line DA, merely adverting, at the outset, that two of the three (those exhibited) must fall on one side of the new line from the point, wherever this be drawn. That the centre cannot be supposed in one of the three lines is immediately obvious, from the proposition referred to. A variety of critical notes upon the Elements will be found at the end of the "Elements of Geometry," by the Editor of this edition of Euclid. Printed by C. ADLARD, Bartholomew Close. BY J. R. 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