ART. XVII.-The First Book of Euclid's Elements. With Alterations and Familiar Notes. Being an attempt to get rid of Axioms altogether; and to establish the Theory of Parallel Lines, without the introduction of any principle not common to other parts of the Elements. By a Member of the University of Cambridge. Third Edition. London. R. Heward. 1830. THIS HIS is an attempt to carry radicalism into Geometry; always meaning by radicalism, the application of sound reason to tracing consequences to their roots, To those who do not happen to be familiar with the facts, it may be useful to be told, that after all the boast of geometricians of possessing an exact science, their science has really been founded on taking for granted a number of propositions under the title of Axioms, some of which were only specimens of slovenly acquiescence in assertion where demonstration might easily have been had, but others were in reality the begging of questions which had quite as much need of demonstration, as the generality of those to which demonstration was applied. In this condition of the science it may be matter of surprise, that no interested supporter of things as they ought not to be, ever bethought himself of appealing to the state of geometry, as evidence of the impossibility of applying rigid demonstration to any thing, and the necessity there is in all human affairs for resting on some assumption or other, which mankind must be taught to believe without proof. The ready defence will be, that the points taken for granted, were what every body knew to be true. The reply to which is, that in the first place, if they were ever so true, no good reason can be given why a thing should not be demonstrated if it can n; and in the next, that the points taken for granted, or some of them, were not such as every body knew to be true, with any thing like that precision of knowledge, which it is the object of science to effect. A common carpenter has a certain sort of knowledge, that if he draws a triangle with two of its sides equal, the two angles or corners opposite to these sides will be found equal also; and nobody has any intention of setting up an assertion that the carpenter is wrong. Why then does the geometrician disturb himself with searching for what he calls a demonstration? and why does not he write down the fact, and give the carpenter as his authority? First, because the carpenter's observation amounts, at best, to a proof of nothing but that in a certain number of instances he found the proposition hold good; but if the geometrician can detect the reason, why the proposition must hold good not only in all the instances which the carpenter did try, but in all that he did not try besides, he has manifestly gained a march upon his rival, and in the space of a few minutes done something vastly more complete, convincing, and satisfactory, than all that has been established on the subject by the rule of thumb, in the process of the carpenter's whole life, and the lives of all possible carpenters to boot. Secondly, because, though there may or may not, be much opening for mistake in this particular observation, yet if a collection of observations of the same description were written down in a book and entitled a treatise on geometry, it is highly probable, or more properly quite certain, that in some or other of them there would be a powerful admixture of error, through the want of a comprehensive view of all the circumstances affecting the result. Thirdly, because the same application of reason which enables the geometer to give the general and universal solution of this particular proposition, will enable him to advance rapidly to the discovery and demonstration of many propositions which all the carpenters in the world would never have dreamt of, and which it would have been utterly in vain to expect that men should ever discover by a merely tentative process. And lastly, because the very exercise and practice of all this, in addition to what may turn out to be the practical value of the discoveries that result, is of high utility from its tendency to throw light on the processes of reasoning, and the conduct of sound argument in general. The question, therefore, of whether a geometer has well or ill done his work, will rest to a considerable extent upon determining, how far he has succeeded in taking his propositions out of their primitive state of assertions found to accord with a limited number of experiments, and placing them in the condition of truths which can be shown to be necessarily applicable to all possible experiments. And this last operation, is what is meant by demonstration. The attempt to get rid of Axioms, is laudable, if successful; but like other rebellions, it must be justified by success. No good ever yet came of axioms. Legitimacy is an axiom; persecution is an axiom. The proposition must be such as was never started on this earth, if it cannot be established by the intervention of an axiom. The modes in which the present attempt has been conducted, may (with the exception of the complicated question on Parallel Lines) be briefly described. Instead of calling on mankind to declare, that they always knew that things equal to the same were equal to one another,' and moreover that magnitudes which coincide with one another are equal to one another,'-they are invited to consider, whether the circumstance that the boundaries would coincide if they could be applied to one another without bar of corporeal substance, or at all events might be made capable of doing so by merely a different arrangement of parts, is not in reality, as respects the objects of geometry, the definition of equality. But if so, this measure of equality is as applicable to three magnitudes at once, as to two; for it is only an act of the imagination in any. If this be true, it removes the First and Eighth Axioms of Euclid at once. It converts the Eighth into a Definition, and the First into a Theorem; and all the intermediate ones, résolve themselves into Corollaries to the First. That the whole is greater than its part,' is omitted as only an identical proposition, that the greatest is greatest.' That two straight lines cannot inclose a space,' is taken, as Euclid really makes it, for the Definition of straight lines. This may be right or wrong; but it is nothing new, for it has been Euclid's for two thousand years. It has been objected to such a definition, that it is only negative.' Is the definition that a straight line is the shortest between its extreme points,' at all less negative?' But the question is not whether any definition is negative, but whether it is good; or in other words, whether it obviously distinguishes the thing in question from all other things, and is easily applicable to the determination of consequences. 6 That all right angles are equal to one another,' is made the subject of a demonstration. There remains then only the question of Parallel Lines; a subject on which so much has been written, and with so little result, as to make it an act of some danger to advance an opinion upon the point. The objection to the existing state of things is, that the Axiom assumed by Euclid, is one which needs demonstration, in at least as great a degree, for example, as the proposition before alluded to, of the equality of the angles at the base of an isosceles triangle. As in that, it is easy to bring evidence that every man who has tried the experiment has found the rule to answer. But, as in that case also, there is still the query, why. There appears no primâ facie reason, why the one proposition should not be as capable of a general demonstration as the other; and if the fact should be, that one of them is not capable of such demonstration, this fact alone would be matter of considerable curiosity, to those who have contracted a taste for such inquiries. The mode in which the_solution is sought, is by endeaVOL. XIII.-Westminster Review. 2 L vouring to demonstrate that if at the extremities of any straight line, two perpendiculars are drawn of equal length and towards the same direction (as may be familiarly represented by three sides of a sheet of paper, or any object of similar form), the straight line joining the other ends of these perpendiculars shall make right angles. And the way in which this conclusion is pursued, is by trying to demonstrate, first, that if the angles at the base or first side are equal, and greater than right angles, the others must be less; and secondly, that if the angles at the base are less, the others must be greater. The demonstration offered of the first of these propositions, is by placing side by side a number of quadrilateral figures equal in all respects to the first, and showing that if their bases are produced, they must successively cut off greater and greater portions of the side of the first or original figure; and conse→ quently, if the number of quadrilateral figures is increased, a time must come when the prolongation of the base of some or other of them, will meet the series of lines formed by the sides of the quadrilateral figures which are opposite to their bases. And as it has been previously established that the side opposite to the base must in each of the quadrilateral figures be parallel to its base, and the angles adjacent to such side be equal to one another-it follows that these angles cannot be right angles, but must be less. The principle appealed to is the same that was produced by M. Legendre in the 7th edition of the Eléments de Géométrie, and withdrawn in consequence of the imperfection of the process by which the remaining step (which, in M. Legendre's case, was that the three angles of a triangle cannot be less than two right angles) was attempted to be established. The procedure for the demonstration of this second part, is by establishing, that if two equal straight lines terminated in the same point, make an angle less than the sum of two right angles, and this be bisected by a straight line of unlimited length which will for distinction be called the axis; and at the outward extremity of each of the two equal straight lines be added another straight line equal to the first, and making with it an angle equal to the first-mentioned angle and on the same side of the line, and so on, lines be added continually; and if the extremities of every two equal straight lines that were added at the same time, be joined by a straight line or chord; each of these chords shall make the angles at the two cusps or corners, where it meets the equal straight lines, equal to one another; and (so long as none of the equal straight lines meets the axis) the several chords shall in succession make greater and greater angles at the cusp, each than the preceding. And the way in which this is proved, is by drawing straight lines from the end of one chord to the end of the next; which shows, almost by inspection, the successive increase of magnitude of the angles at the cusps. A Scholium is added to warn the reader, against supposing that the proof that the angle will continually increase, is any proof that it will attain to a given specified magnitude; a snare into which many of the searchers after a theory of Parallel Lines have notoriously fallen. The next object of proof, is that if in a series of straight lines like the last, the angle at the cusp ever becomes equal to, or greater than, half the angle made by the two first of the equal straight lines, the angular points must lie in the circumference of a circle, whose centre is in the axis, in the part of it which is cut off by the chord; and the series, being continued, must at length meet the axis. And this is done by drawing a line from the angular point of the cusp, so as to make with the last of the equal straight lines that was added, an angle equal to half the angle above described; and showing that the point in which this line cuts the axis, must be equidistant from all the angular points. The next step is, that in a series of straight lines as before, if a straight line of unlimited length both ways, be moved along the axis, keeping ever at right angles to it, such straight line cannot quit or cease to meet the series, without the series having previously met the axis. And this is supported by showing, that when this straight line arrives at any of the cusps, there must always be another pair of straight lines ready for it to pass over, unless in the event of these straight lines having ceased to make an angle with the chord on the side which is towards the axis; and that before this can take place, the angle at the cusp must have been of that magnitude, which has been shown to insure the series meeting the axis. These preparatory propositions are followed by the decisive. one, that in a quadrilateral figure as before described, if the angles at the base are less than right angles, the others are greater. And this is shown by placing a number of the figures in question side by side, prolonging a side of one of the central ones for an axis, and supposing a straight line of unlimited length to move from the vertex along the axis at right angles to it, till it has passed the extremity of the side of the quadrilateral figure which was prolonged to make the axis. If after this it is further moved forward, it must do one of three things;it must either fall in with some of the angular points of the series formed by the bases of the quadrilateral figures, and make an angle at the cusp less than one of the angles |