coincident lines. In both cases, a reference to the idea of direction is made to settle the question. We ought to add, that this axiom, too, is to be found in Proclus,* who tells us, that Possidonius defines parallels " as such that neither recede nor approach each other in the same plane." The same definition has been adopted by Wolfius,† Boscovich,‡ Lacaille, T. Simpson, and Bonnycastle. But it is quite as much an assumption that straight lines do not change their distance as that they do. And thus this definition embodies a property which can only be predicated to belong to the straight line by a consideration of that line itself. 3. After what we have said, it will be superfluous to detail all the axioms which have been substituted in place of Euclid's. We do not deny the possibility of simplifying the theory, either by presenting the mind with a more evident property of continuous direction, or by familiarizing us to it by a multiplication of properties in every variety of form. What we contend for is the fact, that not one of these properties is more derivable from the axiom of straightness than another. Each and all are drawn immediately from a contemplation of rectitude itself. And this observation applies, not only to those assumptions which have been designated as axioms, but likewise to all such as have been tacitly made in en‐ deavours to effect a demonstration. It applies to the processes of Simson, § Creswell,|| Professor Thomson, and the like, who lay down an axiom; but it no less applies * On Defin. last. Elementa univ. math. i. 13. Elem. Math. Univ. § Euc. Notes to Prop. 29, to that of Franceschini,* Exley,† Leslie, and Playfair, who do not. In one and all, reference is made to the nature of a line. But I will not exhaust your patience on this subject. Let me refer you to Camerer's Euclid, Playfair's Do., or Colonel Thompson's Geometry without axioms, for a full examination of the different assumptions and hypotheses which have been made. I ought to add, that in a subsequent little work, entitled "The Theory of Parallels looked for in the Properties of the Equiangular Spiral," the ingenious author last-mentioned abandons his deduction of the second property of a straight line from the first. Before I conclude, it is right that I should add a few remarks relative to the metaphysical connection between the two axioms. Bearing in mind that the one owes its existence to the contact of two lines, and the other to their want of contact, we shall easily perceive that the proper way to go about establishing a connection between them is to aim at effecting a gradual transition from the one to the other. This process, which involves the idea of limits, does not properly belong to elementary geometry, but at the same time it is so intimately connected with our present subject, that we can hardly pass it over without comment. Two proofs of this kind have been given, the one by M. Bertrand of Geneva, the other by M. Legendre. The former‡ is based on the * Opuscoli Math. iii., where the thing to be proved is virtually assumed in Cor. 2. Th. 3; of which the author is perfectly aware when he endeavours, by the method of limits, to establish it in his appendix. Exley, Theory of Parallels. See Develey, Elem. de Géom. ; and Lacroix Do. proof, that the area contained between two straight lines which cut one another at ever so small an angle is greater than the area contained by the two parallels and their intersecting line. But, unfortunately, this is not true at a finite distance from the intersecting point; or, if so, its truth cannot be made plain, so that the demonstration can hardly be said to be legitimate even on its own principles. The other demonstration is given by Legendre.* The following is the process: It is proved that the three angles of a triangle are equal to two right angles by making out, 1. That the three angles of one triangle are equal to those of another (certain one). 2. That this latter, by pursuing a similar process, may be varied, until we approach as nearly as we please to three coincident straight lines. Consequently the angles may be made to differ as little as we please from one angle, viz. that which is contained by two coincident right lines. But this angle is two right angles. Hence the angles of every triangle are equal to two right angles. This, to my mind, is the most satisfactory proof that has ever been offered. The objections to it, however, are fatal as a branch of elementary geometry. Another demonstration was given by Legendre, † who appears to have pushed the subject to its extreme limits, and to have left nothing to be desired. From its analytical character, it is unsuited to this place. The same author subsequently made his demonstration geometrical.‡ In whichever way you view it, there is necessarily made the assump * Géométrie, 12e Ed. Prop. 19. Ibid. Note 2. See also Brewster's translation. Mém. de l'Instit. Vol. xii. See Penny Cyclopædia, Art. Parallel. tion that the magnitude of an angle opposite to a given side of a triangle does not, cæteris paribus, depend on the length of the side. In the analytical method, this assumption may enter into the supposed determination of an equation between a side and the angles; in the geometrical solution it is more palpable. I must apologize for the brief way in which my limits allow me to discuss this matter. LECTURE IV. ON RATIO AND PROPORTION. Measure-ratio-incommensurable magnitudes-proportion-Euclid's doctrine how developed-arithmetical ideas-lead to a common property-that property made the definition-necessity for the sole dependence of the results on the conditions in order that an hypothesis and its conclusion may be reciprocal-propriety of Euclid's definition-its application-definition of inequality of ratios. HAVING explained the principles of the first method of elementary demonstration-Congruity, I proceed to set before you, as briefly as is consistent with clearness, those of the second-the method of Proportionals. Whilst it will be my chief endeavour to offer such simple explanations as may enable you to understand the doctrine, I shall occasionally digress to take notice of objections which have been raised against it, and by pointing out the fallacies they involve, I may be so fortunate as to anticipate difficulties in your own minds. We have hitherto been treating of equality and inequality as manifested by coincidence, excess or defect; we have now to reflect on the same affections as manifested by coincidence, excess, or defect of multiples. This introduces number into our arguments, as the representation of the amount of multiplication. Nor is this the only way in which number is mixed up with geometric researches. A very important branch of geometry is based on the expression of the lengths of lines by the |