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of c to d. The fame may be fhewn, if K is less than C; there Book I. fore, in every cafe there are conditions neceffary to the truth of Torelli's propofition, which he does not take into account, and which, as is eafily fhewn, do not belong to the magnitudes to which he applies it.

In confequence of this, the conclufion which he meant to establish respecting the circle, falls entirely to the ground, and with it the general inference aimed against the modern analyfis.

It will not, I hope, be imagined, that I have taken notice of these circumftances with any defign to leffen the reputation of the learned Italian, who has in fo many respects deserved well of the mathematical sciences, or to detract from the value of a pofthumous work, which, by its elegance and correctnefs, does fo much honour to the English editors. But I would warn the student against that spirit of party, which feeks to introduce itself even into the investigations of geometry, and to perfuade us, that elegance, and even truth, are qualities poffeffed exclusively by the ancient methods of demonstration: The high tone in which Torelli cenfures the modern mathematics, is impofing, as it is affumed by one who had studied the writings of Archimedes with uncommon diligence. His errors are on that account the more dangerous, and require to be the more carefully pointed out.

PROP. IX.

This enunciation is the fame with that of the third of the Dimenfio Circuli of Archimedes; but the demonftration is different, though it proceeds, like that of the Greek geometer, by the continual bisection of the 6th part of the circumference.

The limits of the circumference are thus affigned; and the method of bringing it about, notwithstanding many quantities are neglected in the arithmetical operations, that the errors fhall in one cafe be all on the fide of defect, and in another all on the fide of excess, (in which I have followed Archimedes), deferves particularly to be obferved, as affording a good introduction to the general methods of approxi mation.

BOOK

Supplement

BOOK II.

DEF. VIII. and PROP. XX.

SOLID angles, which are defined here in the fame manner as in Euclid, are magnitudes of a very peculiar kind, and are particularly to be remarked for not admitting of that accurate comparifon, one with another, which is common in the other fubjects of geometrical inveftigation. It cannot, for example, be faid of one folid angle, that it is the half, or the double of another folid angle, nor did any geometer ever think of propofing the problem of bifecting a given folid angle. In a word, no multiple or fub-multiple of fuch an angle can be taken, and we have no way of expounding, even in the fimpleft cafes, the ratio which one of them bears to another.

In this refpect, therefore, a folid angle differs from every other magnitude that is the fubject of mathematical reafoning, all of which have this common property, that multiples and fub multiples of them may be found. It is not our business here to inquire into the reason of this anomaly, but it is plain, that on account of it, our knowledge of the nature and the properties of fuch angles can never be very far extended, and that our reafonings concerning them must be chiefly confined to the relations of the plane angles, by which they are contained. One of the most remarkable of thofe relations is that which is demonftrated in the 21ft of this Book, and which is, that all the plane angles which contain any folid angle muft together be less than four right angles. This propofition is the 21ft of the 11th of Euclid.

This propofition, however, is fubject to a restriction in certain cafes, which, I believe, was firft obferved by M. le Sage of Geneva, in a communication to the Academy of Sciences.

of

of Paris in 1756. When the fection of the pyramid formed Book II. by the planes that contain the folid angle, is a figure that has none of its angles exterior, fuch as a triangle, a parallelogram, &c. the truth of the propofition juft enunciated cannot be questioned. But, when the aforefaid fection is a figure like that which is annexed, viz.

B

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ABDC, having fome of its
angles, fuch as BDC, exte-
rior, or, as they are fome-
times called, re-entering an-
gles, the propofition is not
neceffarily true; and it is
plain, that in fuch cafes the
demonstration which we have
given, and which is the fame
with Euclid's, will no longer
apply. Indeed, it were eafy to fhew, that on bases of this
kind, by multiplying the number of fides, folid angles may
be formed, fuch that the plane angles which contain them
fhall exceed four right angles by any quantity affigned. An
illustration of this from the properties of the fphere is per-
haps the fimpleft of all others. Suppofe that, on the furface
of a hemifphere there is described a figure, bounded by any
number of arches of great circles making angles with one a-
nother, on oppofite fides alternately, the plane angles at the
centre of the sphere that stand on these arches may evidently
exceed four right angles, and that, too, by multiplying and
extending the arches, in any affigned ratio. Now, thefe plane
angles contain a folid angle at the centre of the fphere, ac-
cording to the definition of a folid angle.

We are to understand the propofition in the text, therefore, to be true only of thofe folid angles in which the inclination of the plane angles are all the fame way, or all directed toward the interior of the figure. To distinguish this class of folid angles from that to which the propofition does not apply, it is perhaps beft to make use of this criterion, that they are fuch, that when any two points whatsoever are taken in the planes that contain the folid angle, the ftraight line joining those points falls wholly within the folid angle; or thus, they are fuch, that a straight line cannot meet the planes which contain them in more than two points. It is thus, too, that I would diftinguifh a plane figure that has none of its angles

exterior,

Supplement exterior, by faying, that it is a rectilineal figure, fuch that a ftraight line cannot meet the boundary of it in more than two points.

We, therefore, distinguish solid angles into two species, one in which the bounding planes can be interfected by a straight line only in two points; and another where the bounding planes may be interfected by a straight line in more than two points: to the first of these the propofition in the text applies, to the second it does not.

Whether Euclid meant entirely to exclude the confideration of figures of the latter kind, in all that he has said of fo. lids, and of folid angles, it is not now easy to determine: It is certain, that his definitions involve no fuch exclufion; and as the introduction of any limitation would confiderably embarrass these definitions, and render them difficult to be understood by a beginner, I have left it out, referving to this place a fuller explanation of the difficulty. I cannot conclude this note without remarking, with the hiftorian of the Academy, that it is extremely fingular, that not one of all those who had read or explained Euclid before M. le Sage, appears to have been fenfible of this mistake. (Mémoires de l'Acad. des Sciences 1756, Hift. p. 77.) A circumstance that renders this still more fingular is, that another mistake of Euclid on the fame fubject, and perhaps of all other geometers, efcaped M. le Sage alfo, and was firft difcovered by Dr Simfon, as will presently appear.

PROP. IV.

THIS very elegant demonftration is from Legendre, and is much easier than that of Euclid.

The demonstration given here of the 6th is alfo greatly fimpler than that of Euclid. It has even an advantage that does not belong to Legendre's, that of requiring no particular conftruction or determination of any one of the lines, but reafoning from properties common to every part of them. This fimplification, when it can be introduced, which, how

ever,

ever, does not appear to be always poffible, is perhaps the Book II. greatest improvement that can be made on an elementary demonstration.

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The problem contained in this propofition, of drawing a ftraight line perpendicular to two straight lines not in the fame plane, is certainly to be accounted elementary, although not given in any book of elementary geometry that I know of before that of Legendre. The folution given here is more fimple than his, or than any other that I have yet met with it alfo leads more eafily, if it be required, to a trigonometrical computation.

BOOK III.

DE F. II. and PRO P. I.

These relate to fimilar and equal folids, a subject on which Book III. mistakes have prevailed not unlike to that which has just beenmentioned. The equality of folids, it is natural to expect, must be proved like the equality of plane figures, by fhewing that they may be made to coincide, or to occupy the fame fpace. But, though it be true that all folids which can be fhewn to coincide are equal and fimilar, yet it does not hold conversely, that all folids which are equal and fimilar can be made to coincide. Though this affertion may appear fomewhat paradoxical, yet the proof of it is extremely fimple.

Let

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