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lines, has made use of an affumption equally unwarrantable with that mentioned above. That the theory of parallel lines, as it is given in the Elements, is very imperfect, cannot be denied; but no one has yet been fubftituted in its place which is not equally defective; and in some instances ftill more exceptionable : particularly as they are founded on a definition which is derived from an adventitious property of thofe lines, instead of one which is inherent and necessary, as the nature of the subject requires.
Whether EUCLID was the author of this axiom cannot perhaps, at this time, be easily determined ; but it is certainly a disgrace to the Elements. The truth of the property here assumed as a thing to be granted, is so far from being obvious, that it requires demonstration as much as any Prop. in the Elements; and it is always observed that learners, instead of giving that ready assent to it which an axiomatical principle requires, receive it with doubt and hesitation, and are scarcely able to comprehend the meaning of it. The one which is here made the 4th postuJate, though nearly the same thing in effect, is much more clear and intelligible.
PROP. I. BOOK I. In the demonstration of this proposition, by EUCLID, that part which relates to the intersection of the circles is, very improperly, omitted; for in a work of this kind, nothing, however evident, ought to be taken for granted; and particularly at the first outset, where a strictness of eluciation is peculiarly necessary. The passing of the circles through each other's centres is, indeed, a sufficient reason why they must cut each other ; but this should certainly have been mentioned in the demonstration.
PROP. 2. BOOK I. PROCLUs, and other writers, have observed, that this problem admits of several cases, according to the situation of the point A ; but there is only one of them that can properly be called a separate case, which is when the point A is at either of the extremities of the given line : and in this case, if a circle be described from the given point, at the distance ce, any of the radii of that circle will be the line required. In all other situations of the point A, whether in the line AB, or out of it, the construction and demonstration will be the same as that given in the text; which differs from Euclid's only in the producing of the line DA, after the circle Fhg is described; this being thought more conformable to the terms of the proposition.
Book I. In the construction of this problem the line at may fall upon the line AB, and then the thing required is done. The given lines c and AB may also meet each other, at the point A; and then a circle described from that point, with the radius c,'will cut off from As the part required. This case occurs in the conftruction of the 5th proposition following, and in several other parts of the Elements, and, for that reason, ought to have been mertioned. In all other positions of the two given lines Euclid's construction and demonstration are general.
PROP. 4. BOOK I. The demonstration of this proposition has been frequently objected againit, as being too mechanical. But this complaint is frivolous and ill founded; for the operation of placing one triangle upon the other, is a mental one, and what is to be considered as possible to be effected, rather than actually done. There is, besides, no other way in which the equality of these figures can be established, fo that any cavils about its merits or defects are entirely precluded.
PROP. 5. BOOK I. EUCLID, in his demonstration of this propofition, has shewn that the angles below the bafe are, also, equal to each other. But as this property is never referred to throughout the Elements, except in the demonstration of the 2d case of the 7th propofition following, the whole of which is both aukward and unnecessary, it would have been better to have omitted it, and confined the demonftration, in the present instance, to the equality of the angles above the base, which is a property much more generally useful.
PROP. 6. Book I.
The demonftration of this propofition, in EUCLID, is immethodical, and defective. It is not sufficient to shew that one fide is not greater than the other, but it ought, also, to be fewn that it is not less, before their equality can be fairly inferred. It is true, indeed, that either of the fides may be taken at pleasure, and the fame thing will follow : but this observation should have been made; and then the premises, which they do not at present, would have authorized the deduction required. The fame objection may be made to several other propositions in the Elements.
PROP. 7. Book I. This is the same as Euclid's 8th proposition, but demonstrated in a different manner, in order that the preceding one, which is altogether useless, might be omitted. PROCLUS demonstrates it in nearly the same manner; but he makes three cases of it, when it may be done generally in one; for if the longest fides, or rather those which are not fhorter than any other, be applied together, there can be no ambiguity in the species of the triangles.
PROP. 8. BOOK I.
This proposition is made an axiom by EUCLÍD; but it is certainly not a truth of that kind : for when two right, angles are found in separate and distinct figures, there is nothing in the definitions or poftulates, from which their equality to each other can be fairly inferred.
It is not fewn by EUCLID, in his demonstration of this problem, that the circle made use of in the construction, will cut the given line in two points, which as much requires proof as Prop. 2. Book III. which is nearly its converse. For this reason the construction given in the text has been preferred; but in a work where the utmost scientific rigour is required, it would be better to construct the problem without the intervention of the circle, by means of right lines only, which may easily be done.
PROP. 13. BOOK I. In the enunciation of this Prop. by EUCLID, the angles are said to be either equal to two right angles, or toge
ther equal to two right angles; but the former part of this seems to be unnecessary; for in all cases, whether the angles be each of them a right angle, or not, they are together equal to two right angles.
PROP. 16. Book I.
As the outward angle of a triangle is afterwards shewn to be equal to the two inward oppofite angles, it is much to be wished that the present Prop. which is only a partial case of the former, could be removed from the Elements : but this cannot easily be done; for the following proposition, and the first relating to parallel lines, are not otherwise to be demonstrated. The next Prop. in EUCLID, is, however, quite unnecessary, as the first place in which it occurs is Prop. 18. B. 3, where a reference may be as readily made to the general propofition.
PROP. 19. BOOK I. The demonstration of this propofition, as it is given by Euclid, is extremely defective; for the whole design of the problem is to shew that of three right lines, under certain specified restrictions, a triangle may be formed; and as no use whatever is made of these restrictions, either in the construction or demonstration, both the arguments adduced, and the conclusions derived from them, are entirely nugatory. This defect was observed by MR. SIMPSON, between whom and his antagonist Dr. SIM. SON, it occafioned some controversy, which drew from the latter some very hafty unfcientific expressions, not much comporting with the character of fo ftri&t and accurate a Geometrician.