Book III. fide. The other cafe is, when the point E falls upon the Foint D, and the third when the point E falls between A and D, and the point F beyond D.-But fince the difference between the Demonftrations of Dr. Gregory's cafe, and that of the other third above-mentioned only confifts in the second axioms being used in his and the third in the other cafe, the Demonftration of the Doctor's cafe might very well ferve for that of the other.-As to the cafe of the point E falling upon D. here [by prop. 34. book I.] the triangle A D C is one half the parallelogram A D. [fee fig. of prop. 35.] On PRO P. II. BOOK II. The construction of prop. 11. book 2. gives that of this other problem. To one given right line to add another right line fuch, that the rectangle contained under the added line and the whole line taken as one line, fhall be equal to the fquare of the whole line. Let A B be the given right line, then [by 46. 1.] describe the fquare ABDC of A B, and [by 10. 1.] divide AC into A two equal parts at E, join BE, produce C A to F. and [by 3. 1.] make E F equal to в E. alfo defcribe CG the fquare of c F. now continue out the given right line A B to meet the fide C F E G B H D K GK of the fquare CG in the point H. then will A B be continued out to H, fo that the rectangle under A H and BH will be equal to the fquare of the given right line A B. The demonftration of this fo little differs from that of prop. 11. book 2. that one may very well ferve for both. On DEF. XI. BOOK III. This definition of the fimilarity of the fegments of circles from the equality of the angles in them, has fomething of cbfcurity in it; for fince it is not yet known whether the angles in them be all equal to one another; this being demonftrated afterwards at prop. 21. of the third book, it causes the definition to be attended with a doubtful idea, and fo its meaning not readily and clearly perceived; for if poffible, no definition fhould be expreffed in words that the the mind is not fully and evidently præ-convinced of their truth. The following definition therefore of fimilar fegments of Circles, viz. that they are those whose bases are in the fame proportion to their diameters, seems to convey a much clearer notion of fimilar segments than that other of Euclids, to thofe who know the meaning of the words in the fame proportion; but fuch a definition as this cannot be put to the third book without a manifeft breach of the order and just method observed all along in these Elements; that nothing is taken as true in any one part of them without it be before established either as a definition, axiom, or demonftrable propofition.-Similar fegments are first mentioned in the third book, at prop. the 23d and 24th; now the 23d propofition is neither clearly worded, nor of any great ufe that I can perceive; for the 24th propofition is demonftrated without it; and the 26th propofition depends upon the 24th; and as I don't find any other mention of this 23d propofition, or use of it, in demonftrating any other in thefe Elements of Euclid, I think it might have as well been left out, and the definition of fimilar fegments too: But if any be retained, it fhould be fuch a definition as I have mentioned above; and this would better have a place amongst thofe of the 6th book, than be in the third book, where it, nor any other, is not wanted at all. Note, the 23d and 24th propofitions of book the 3d, may be thus expreffed in one propofition: If the segments of equal circles be equal, their bases will be equal; and on the contrary, if their bafes be equal, the fegments will be equal; for fince the equality of magnitudes proceeds from their total congruency, when they are properly applied to, or laid upon one another; if the two equal circles be arightly applied, they will coincide or agree in every part; wherefore it evidently follows, that two equal, and the felf fame fegments of those equal circles thus applied to one another muft neceffarily have one and the fame right line for their bafe; and if they have one right line for their base, the fegments will be equal.-What demonftration can make. the truth of this more convincing, it may indeed be nearly as much taken for an axiom, that equal circles have equal fegments, when their bafes are equal, as is the first definition of the third book, that equal circles are thofe whofe diameters, or femidiameters are equal. Euclid's demonftrations of the 23d and 24th propofitions, never without gave me a stronger conviction of their truth, than I had without any demonftration at all; however, I fhall not contend this with those that think otherwife, the matter is of no great moment. On PROP. XII. BOOK III. The figure of this Propofition is not fo well adapted to affift the imagination as it might be; the fictitious cen B D F G A ters F, G of the circles A B C, ADE being affumed too remote from the middle points or real centres of them; this offends the fight and evidence of fenfe; and is apt in fome degree, to obfcure and leffen the clear apprehenfion and evidence of the demonstration. I therefore think the annexed figure will do better than that of the Propofitions in the third book. B E A H G D This propofition, as well as the 12th, may be inverted thus. If two circles touch one another, a right line drawn from the centre of one of them through the point of con tact, will pass through the centre of the other circle. Let there be two circles B AC, DAE whofe centers are F and G touching one another at the point A. I fay the right line FAH drawn from the center F of the circle BA C, will likewise pass through the center G of the circle ADE. If it does not, join the centers F and G by the right line FG, then [by prop. 12.] the right line F G will pass through the point of contact A, as well as the right line F AH, and fo two different right lines will pafs through the fame two points F, A, which is impoffible. On PROP. XIII. F C BOOK III. firft part of this propoElements; but I think I have feen two figures of the fition, in a late edition of Euclid's there is no occafion for two; the truth of the propofition is felf evident, and were it not for encreafing the number of of axioms, it might have paffed among these without any demonstration at all.-I believe no perfon of tolerable apprehenfion, upon reading the enunciation of this propofition, ever doubted its truth, or wanted any demonftration to convince him of its truth.-The demonstration here is only for forms fake. And the second part of the propofition, where the circles touch one another outwardly, feems less to want a demonftration; there is no man that can conceive how two circles that touch one another outwardly, fhould do fo in more points than one; and the figure of this part of the proposition in the third book is very ill fuited to affist the imagination at all in this matter. Qn PROP. XVI. BOOK III. It is faid in the enunciation of this propofition, "That << no other right line can be drawn between that fame "line [i. e. the tangent] and the circumference of the "circle."I have obferved beginners always hefitate here, and when I first read this paffage, many years ago, I took it to be wrong.—I thought many right lines might be drawn between the circumference of a circle and its tangent; and fo indeed there may; but then thefe are fuch right lines as do not iffue from the point of contact, which Euclid fuppofes them all to do; but has not exprefsly faid fo, which he fhould have done, and then all would have been aright. Hence inftead of the words above, it should have been faid, "No other right line can be drawn from the "end of that diameter but what will fall within the circle, "and not between the circumference and that right line.” On PROP. XX. BOOK III. In a late edition of Euclid's Elements, fault is found with Dr. Gregory's tranflation of the first words of the fecond part of the demonftration of this propofition, it is "rurfus inclinetur," when it should have been "rurfus "inflectatur." But I think Dr. Gregory did this on purpofe to avoid a new definition of a strait line being inflected to a ftrait or curve line, which is rather an optical than a geométrical definition. This propofition appears to me not to be well expreffed, because, according to the common enunciation, it will not be Book III. be not true in general: for instance, the angle BEC is an angle at the centre of the circle B ACG, and the angle BAC is an angle at the circumference, both being upon the fame bafe BC, or part of the circumference; but the angle B E C is not double of the angle B AC; therefore I think the 20th propofition fhould be thus enunciated. "The angle at the center of a circle is the double of the angle in a greater "fegment, when the angle at the center is formed by the two femidiameters drawn to the extremity of the bafe "of the fegment." There is another propofition of this fort, when the angle B A C is in a leffer fegment of a circle, viz." the angle BEC at the center is double to the angle "BAF, being the compliment of the angle BAC of the "leffer fegment, to two right angles, when the angle of "the leffer fegment has for its base the fame part of the "circumference as has the other angle at the center." Draw any two right lines BG, CG, to any point G in the circumference of the greater fegment of the circle. Then [by prop. 20. 3.] the angle BEC at the center B F C E G angles BAC, BAF [by 13 1.] take the common angle BAC from both, and the angle B G C will be equal to BAF. But BEC is double to B G C, therefore BEC is double to BAF. On PROP. XXXV. BOOK III. This propofition has four cafes, although there are but two mentioned in most of the editions of Euclid's Elements, except in the translation of Campanus from the Arabick, were they are all to be found. The cases in this Euclid of Dr. Gregory's, are the most easy, requiring no demonftration at all, viz. when the point E wherein the two right lines AC cutting one another fall in the center of the circle, and the most difficult when neither of the |
