formed the most remote conception, and their most ingenious speculations penetrate but a small way into that vast field of knowledge which has been laid open by this admirable instrument of analysis. By its successful application also wonderful improvements have been made in many departments of natural philosophy, and particularly in the interesting investigations of astronomy, which without its powerful aid could never have attained to its present high state of perfection. Problems of peculiar difficulty which could not be solved by the ancient methods of demonstration, or which admitted of solutions so intricate and tedious as to require no common exercise of patient attention to peruse and understand them, are by the modern calculus solved with the greatest expedition and facility, and it is probable that its sphere of action may be yet greatly extended.

No part of elementary geometry since the time of Euclid hascaused so much perplexity and annoyance to geometers as the subject of parallel lines. There is no great difficulty in demonstrating, that if two straight lines make equal alternate angles with any third straight line which either meets or intersects them, the two lines are parallel; but to demonstrate the converse of this proposition, that if a straight line fall upon two parallel straight lines they will make equal alternate angles with it, is a feat which has never been hitherto accomplished in a manner entirely unexceptionable. Various reasons have been assigned for this failure, such as, the imperfection of our definition of a straight line, the idea of infinity entering into the commonly received definition of parallel lines, and a few others which do not on consideration seem to give a satisfactory account of it. Euclid would no doubt endeavour

at first to demonstrate the properties of parallel lines by the aid of the truths which he had already established, but, finding these insufficient, he was under the necessity of assuming an additional principle in order to effect his purpose. This principle forms his twelfth axiom, a proposition which, although true, is by no means self-evident, for the ingenuity of modern geometers has frequently been employed in endeavouring to furnish a faultless demonstration of it. Some have proposed to remove the difficulty, by introducing an axiom more obvious than that of Euclid, and others by a new definition of parallel lines. Those, however, who have adopted either of these methods have encountered the same or a similar obstacle; and in almost every attempt hitherto made to establish the properties of parallels in a manner purely geometrical, some unwarrantable element has, on close scrutiny, been found to have been unwarily introduced equivalent to an assumption of the very truth to be demonstrated. Geometers, therefore, are now generally disposed to regard every attempt to establish the properties of parallel lines geometrically as a hopeless task, in the accomplishment of which all who engage will meet with certain disappointment. The majority of geometers have hitherto directed their attention solely to the lines themselves, and treated the question as one of lineal magnitude only. On consulting the 29th proposition of the first book, and the demonstration of it which Euclid has given, there is nothing. doubtful with regard to the nature and position of the lines; they are given straight and they are given parallel; and the only question proposed is to determine the angles which they make. with any straight line which falls upon them. It appears therefore to be a question of angular rather than of lineal

magnitude; and if a question of angular magnitude, it must necessarily have an intimate connexion with the important truth, that the three angles of every triangle are together equal to two right angles,—a proposition on which the determination of all kinds of angular magnitude depends. By the method of supraposition, this proposition cannot be established until the properties of parallel lines have been demonstrated, and this cannot be effected without the assumption of some additional principle equivalent to Euclid's twelfth axiom. Again, on the supposition that it is a question of angular magnitude, it is absolutely requisite that the same proposition should be proved independently of parallel lines, before a legitimate demonstration of their properties can be obtained. Such a proof, however, of this leading truth has never yet been given which is entirely satisfactory, nor can be given on the principle of supraposition alone. Such is the position of difficulty in which geometers have been placed in the discussion of this subject, and from which they have hitherto found no effectual means of escape. Since the principle of supraposition alone, then, is insufficient to free the subject from difficulty, and enable geometers to attain their object: if a satisfactory solution of the question is to be obtained, recourse must be had to some other mode of demonstration.

It is also worthy of observation, that those geometers who have introduced the idea of motion into their demonstrations have been most successful in demonstrating the truth of Euclid's twelfth axiom. Professors Playfair and Thomson of Glasgow, by the introduction of this element, have been able to prove, independently of parallel lines, that the three angles of every triangle are together equal to two right angles, and

thus appear to have made a step in advance towards the solution of the difficulty. The methods they employed are indeed rather too mechanical, but they seem to have been groping their way to the principle of solution, although they had not reached it. As the method of determining the amount of angular magnitude contained in the angles of any figure must be very different from that of ascertaining the quantity of space which it encloses, it is by no means surprising that it depends on a mode of proof very different from that by which the equality or inequality of rectilineal figures is established. Supraposition may be fully competent to attain the latter object, but it is found to be altogether incapable of accomplishing the former, or of estimating the amount of angular magnitude contained in the angles of the simplest rectilineal figure. The introduction therefore, in a plain and direct manner, of the principle of variation, the other mode of demonstration by which such wonderful discoveries have been made in modern times, appears to be absolutely necessary; and this necessity is even. tacitly and unintentionally acknowledged by those who have adopted a different plan. This has been done particularly by Legendre, who has demonstrated that the three angles of a triangle are equal to two right angles, by an indirect introduction of the principle of variation. He first proves that the three angles of one triangle are equal to those of another having an angle of a particular description. He then constructs a second triangle on the same model, then a third, and, by continuing the series of triangles far enough, the three sides are made to approach nearer than by any assignable difference to three coincident straight lines. The three angles must at the same time approach nearer than by any con

ceivable difference to one angle, viz., that which is made by two straight lines which are in one and the same straight line. Now this angle is two right angles. The three angles of every triangle are therefore equal to two right angles. By this method of introducing the principle of variation an approximation only to the truth is attained, whereas, if the same principle were applied directly, the conclusion might be arrived at with perfect accuracy. A line might be drawn from the vertex of any triangle to the base, and by a very simple application of this principle the vertex of the triangle might be conceived to move along this line, gradually changing the form of the triangle, and collecting the three angles into one. When the vertex reached the base, the vertical angle would then become equal to the sum of the three angles, and this sum, when estimated by a measure of angular magnitude formerly determined, would be found exactly equal to two right angles. Then, by means of the definition of parallel lines here adopted, the proof that lines equally inclined to one line must be so to every other line, follows directly from the proposition already stated, and thus the theory of parallel lines is freed from every difficulty. The reason why the ancient geometers failed in their attempts to demonstrate the properties of parallel lines now becomes apparent. They were unacquainted with the principle on which the proof of these properties depends. They employed supraposition alone, and on this principle as a solid basis they endeavoured to rear the science, and, to their honour it must be acknowledged, that they have executed their task in a manner highly creditable to their mental powers and persevering industry. Their nearest approach to the principle of variation is seen in their method