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follows:-Through a given point only one parallel can be drawn to a given straight line. Our author's objection to Euclid is :Of this, where is the proof? The question is now reduced in our minds to the following:-Is the axiom of our author more evident than that of Euclid? (we here appeal to our senses): next, is it so much more evident that it is worth while to append to Euclid six propositions and seventeen pages of matter? We answer no to the first, and of course to the second. If a proof could be supplied of that which at present is an axiom, the rest might stand: we say might, for we have not thought it necessary to examine further; but up to the present time, we consider that no advantage has been gained over Euclid.
Our author has not said a word on the case in which the angle CAB is so taken that BD falls below BC. The angle BAC is, by hypothesis, any angle less than the sum of two right angles. Let the reader take an angle less than sixty degrees, and try to construct the figure, and follow the demonstration; let him then do the same with Prop. xxviii. D, and he will see that neither of them are general. In order to be certain that ABD is greater than ABC, the hypothesis, that BAC is less than two right angles, must be increased by adding that BAC is greater than one right angle. The angle ABC, for anything yet shown to the contrary, may with CAB make up anything short of two right angles, that is, for BD to fall in the way drawn in the figure, BAC or ABD must be greater than ABC, that is, must be greater than a right angle; for, anything previously proved notwithstanding, if BAC be less than one right angle, ABC may be greater.
Our author endeavours to prove several of the other axioms. If his proofs were called illustrations or exemplifications, we should not object to them; but as it is, we are at a loss to see in what the proof consists. Apollonius of Perga, according to Proclus, had preceded him in this attempt: we give the demonstration by Apollonius of the axiom, that things which are equal to the same are equal to one another. He argues, that if A is equal to B, it occupies (may be made to occupy) the same place as B. And if B is equal to C, it occupies the same place as C, whence A and C оссиру the same place, or are equal: on which Proclus remarks, that the position, things which occupy the same place are equal,' and its converse, are not more evident than the thing which is to be proved. Our author proceeds as follows (page 4): The establishment of some universal proposition is called a demonstration. A consequence is, a conclusion, the truth of which is shown to be so connected with the truth of some preceding position or statement, that the preceding cannot be true, with
out the other being true also.' Then (page 7), having drawn from definition the conclusion that A and C, if both equal to B, can be made to coincide in boundary with B, our author remarks, without reference to any proof, But because A and B would each coincide with C; if the boundaries of both could be applied to those of C at once, A and B would coincide with one another,' whence he infers the equality of A and B. Is not the clause just quoted precisely the axiom of Euclid? The latter says, things which are equal to the same are equal to one another,' and looking at the definition of the term equality, (which we admit to be improperly placed among the axioms,) we find things which fit on one another (rà QaρμÓGOVTα ET' λλnλ) are equal.' To fit on one another can mean nothing but to coincide in boundary. Substituting, therefore, for the word equal its definition, the axiom of Euclid takes the following form, opposite to which we write the assumption of our author.
We are at a loss to see what advantage is gained by our author in the way of proof.
An angle, says our author (Preface, page x.), or the thing talked about under that name, is, whether geometers know it or not, a plane surface. His is the first work, which we know, in which this idea is fairly brought before the beginner. We suspect he is quite right; and that in the extension of the term equal to unlimited figures which coincide in all their parts, as well as to limited figures, will be found the ultimate resting point of the theory of parallels. Our author's definition of an angle is the plane surface (of unlimited extent in some directions, but limited in others) passed over by the radius vector in travelling from one of two divergent straight lines to the other.' Had our author stuck close to his definition, the
demonstration of Euclid's axiom, given by M. Bertrand, ought to have been sufficient; but in arguing against that demonstration (page 147), he observes, that all references to the equality of magnitude of infinite areas are intrinsically paralogisms. This astonished us not a little when compared with his own definition of an angle, for we could not suspect our author of playing upon the difference between the words ́unlimited' and infinite.' On further inspection, however, we found that the definition was a dead letter, and that our author's treatment of the angle was precisely that of Euclid. We wonder, therefore, that the definition should have been inserted; for it is in the definition only, and the difficulty which a beginner must find in settling his ideas of greater, less, and equal, on that definition, that the whole objection to M. Bertrand's demonstration turns.
We have hitherto omitted all mention of a curious and novel part of the work, which, though liable to objection, has the stamp of talent upon it in no ordinary degree. The author calls it an intercalary book, to be omitted until the student is, in some degree, familiar with geometry. We also should recommend the beginner to leap it, if he would ever hope to make any progress. It is intended to supersede the definitions and axioms relating to the straight line and plane, by generating the first (to which we will confine ourselves) from the point of contact of two spherical surfaces, one of which decreases and the other increases, the centres remaining fixed. This is, we certainly admit, to begin gemino ab ovo. Our author's description of it is as follows; we prefix some new definitions which he requires :
called a body.
Any thing that can be made the object of touch is
A body whose particles are immoveable among themselves, at least by any force there is question of employing, is called a hard body.
(Preface, page vi.)- A solid may be described-all the points in whose surface shall be equidistant from a given point within; such a solid is called a sphere. A sphere may be turned in any manner whatsoever about its centre, without change of place. Consequences deducible from this are, that if two spheres touch one another externally, they touch only in a point; and if they are turned as one body about the two centres, which remain at rest, the point of contact remains unmoved. Hence, if about two assigned points be described a succession of spheres, touching one another, any number of intermediate points may be determined that shall be desired, which, on the whole being turned about the two centres, shall be without change of place; and if this be extended to imagin* See the Society's Treatise on the Study of Mathematics, page 78.
ing one sphere to increase continuously in magnitude, and the other to decrease, the line described by their point of contact will be without change of place throughout: such a line is called a straight line.'
The above is sufficient to give a notion of the scope of the book; the idea of hardness involved is not meant to include impenetrability-two such hard surfaces may intersect. Four points are said to be equidistant, two and two, without reference to the length contained between them, when the first two can, by change of place only, be made to coincide with the second. We think we see in this same hardness and its consequences something very like an axiomatic distinction between absolute and relative position; want of room obliges us not to dwell on this part, and to avoid the imputation of just hinting a fault, we avow that we really do not know whether we could establish that point or not. We proceed to shew what we
conceive to be a concealed axiom.
The pinch and nip' (we thank the author for teaching us those words) of the intercalary book lies in the demonstration of Proposition V. (page 16): if two spheres touch one another externally, they touch only in a point.' It has previously been fairly deduced from the assumed definitions, that a sphere does not suffer change of place by any motion round its centre; that is, any point of absolute space, which either is or is not occupied by a point of the sphere in one of its positions, is in the same predicament for every other position. The author then gives his demonstration of the several cases; namely, that the two spheres neither meet in a surface, a closed line, an open line, nor a plurality of insulated points. Hence he infers they can meet only in one point; but the demonstration of each of the cases requires that the intersection supposed in the reductio ad absurdum should, by some motion of the sphere, suffer change of place, or be capable of being moved with one of the spheres, so that some absolute points of either sphere, which were parcel of their intersection in one position, are not so in another. But what if the following proposition should happen to be true? Not only does the whole sphere not suffer change of place by any motion round its centre, but upon the sphere a line or zone can be drawn, which also shall not suffer change of place.' We are sure that the author does not get through his demonstration without a tacit denial of the preceding, which it is not very obvious how to refute, even to a moderately well informed mathematician. Can any one say when he first comes to consider the subject, that something might not be drawn on the sphere which should have the preceding property? But the question is not even this; for, supposing that we are justified, on the evidence of our senses, in OCT., 1833-JAN., 1834.
rejecting the possibility of the preceding hypothesis, the question arises, do we, in such rejection, say that which is more evident to the senses than that two straight lines cannot inclose a space?' For the very object of the intercalary book is to supply geometrical evidence of the last-mentioned axiom. But we have another objection to the preceding proposition. The hypothesis that the spheres touch externally' is not even alluded to in the demonstration. The author usually cites his hypothesis by a marginal reference; but certainly not in the margin, nor, so far as we can see, in the text, does the limitation implied in the hypothesis affect the argument. Unless, therefore, we are wrong on this point, the proposition proves not only that contact, but all species of intersection, is limited to a single point. Even the author himself appears to have been unconsciously led by the obvious tenor of his argument, for the word touch' gives place to coincide' throughout the proposition, except only in the preliminary and concluding enunciations. The author's words are, if this,' namely, that they touch in more than a single point, be disputed, let it be assumed that they coincide, &c.' But we naturally look for, let them touch in more points than one,' and for some distinction between touching' and coinciding;' coincidence is afterwards shown to be possible, though the result of every case of this proposition is, that the coincidence assumed for argument's sake is announced to be impossible.
The author conjectures that Napoleon had the idea in his mind, that in the properties of the circle, or still more probably of the sphere, might be discovered the elements of
geometrical organization.' We think it more likely that the idea in the mind of that eminent practical geometer' was, that in the possession of Europe, or still more probably of the sphere, might be discovered the elements of satisfied ambition, at least for animals with lungs and without wings. But we think the author has got the wrong end of a story about Mascheroni's Géométrie du Compas, not much known, which, as every thing relating to Napoleon has its interest, we will, therefore, cite with abridgment from the Preface to M. Carette's translation of that work:
Mascheroni published his Géométrie du Compas at Pavia in 1797, towards the end of Napoleon's stay in Italy. The latter had several conversations with the author about his work, and when he returned to France was invited by Français de Neufchateau to meet a large party of members of the Institute. Laplace and Lagrange were there, and Bonaparte, in conversation with them, but particularly with the former, made known the work of Mascheroni, for the first time in France, and showed the solution of some of the