In equal circles, the angles have the fame ratio to one another as the circumferences on which they stand, whether they ftand at the center or at the circumference. And befides alfo the fectors have the fame proportion because they stand at the centers. Let ABC, DEF be equal circles; and let the angles BGC, EHF be at their centers G, H: and the angles BAC, EDF at the cir cumferences: I fay that it is, as the circumference BC is to the circumference EF fo is both the angle BGC to the angle EHF and the angle BAC to the angle EDF: and besides so is the sector GBC to the fector HEF. Take any number of contiguous circumferences (by 1. 4.) CK, KL equal to the circumference BC, and alfo any number of circumferences which may accidentally happen FM, MN equal to the circumference EF; and let GK, GL; HM, HN be joined A B C R F H Book VI. Wherefore because the circumferences BC, CK, KL are equal to one another; also (by 27.3 the angles BGC, CGK, KGL are equal to one another: whatfoever multiple therefore the circumference BL is of the circumference BC; the fame multiple alfo is the angle BGL of the angle BGC. Certainly for the fame reafon also, whatsoever multiple the circumference EN is of the circumference EF; the fame multiple alfo is the angle EHN of the angle EHF: and if the circumference BL be equal to the circumference EN; alfo (by 27. 3.) the angle BGL is equal to the angle EHN and if the circumference BL be greater than the circumference EN; alfo the angle BGL is greater than the angle EHN; and if lefs, lefs: there being four magnitudes; the two circumferences BC, EF; and the two angles BGC, EHF; there have been taken the circumference BL and the angle BGL equimultiples of the circumference BC and of the angle BGC; and the circumference EN and the angle EHN any other equimultiples which may accidentally happen of the circumference EF and of the angle EHF; and it has been demonftrated that if the circumference BL exceed Book VI. exceed the circumference EN also the angle BGL exceeds the angle EHN: and if equal, equal: and if lefs, less: it is therefore (by 5. def. 5.) as the circumference BC is to the circumference EF fo is the angle BGC to the angle EHF: but as the angle BGC is to the angle EHF fo is (by 15. 5.) the angle BAC to the angle EDF; for they are (by 20. 3.) the doubles, each of each: therefore as the circumference BC is to the circumference EF fo is both the angle BGC to the angle EHF; and the angle BAC to the angle EDF. Wherefore in equal circles, the angles have the fame ratio to one another as the circumferences on which they stand; whether they stand at the centers or at the circumferences. Which was to be demonstrated. I fay alfo, that as the circumference BC is to the circumference EF fo is the sector GBC to the sector HEF. For let BC, CK be joined; and the points X, O being taken in the circumferences BC, CK; let BX, XC; CO, OK be joined. And because the two BG, GC are equal to the A B X C F D rence BC is equal to the circumference CK; also the remaining circumference, which makes up the whole circle ABC, is equal to the remaining circumference which makes up the fame circle; fo that alfo (by 27. 3.) the angle BXC is equal to the angle COK; wherefore (by 11. def. 3.) the segment BXC is fimilar to the fegment COK; and they are upon equal straight lines BC, CK : but fimilar segments of circles upon equal straight lines (by 24. 3.) are equal to one another: therefore the fegment BXC is equal to the fegment COK but the triangle BGC is also equal to the triangle CGK; therefore the whole sector GBC is equal to the whole sector GCK. Certainly for the fame reason also the sector GKL is equal to either GKC or GCB: therefore the three fectors GBC, GCK, GKL are equal to one another. Certainly Certainly for the same reason also, the sectors HEF, HFM, HMN are equal to one another; whatsoever multiple therefore the circumference BL is of the circumference BC; the sector GBL is also the same multiple of the sector GBC. Certainly for the fame reason also, whatsoever multiple the circumference EN is of the circumference EF; the fector HEN is alfo the fame multiple of the sector HEF: And if the circumference BL be equal to the circumference EN; alfo the sector GBL is equal to the fector HEN: and if the circumference BL exceed the circumference EN; the sector GBL alfo exceeds the sector HEN; and if the one is deficient, the other is deficient. There being then four magnitudes, the two circumferences BC, EF; and the two fectors GBC, HEF; and equimultiples have been taken of the circumference BC, and of the fector GBC, viz. the circumference BL and the sector GBL; and equimultiples also of the circumference EF, and of the sector HEF, viz. the circumference EN and the sector HEN and it hath been demonftrated, that if the circumference BL exceed the circumference EN, the fector GBL alfo exceeds the fector HEN; and if equal, equal; and if the one is deficient the other is deficient. Therefore it is (by 5. def. 5.) as the circumference BC is to the circumference EF fo is the sector GBC to the fector HEF." Cor. And it is manifeft (by 11.5.) also, that as the sector is to the sector fo is the angle to the angle. THE END OF THE SIXTH BOOK. VOL. I. * I THE THE CONCLUSION. A FTER the first fix books of these elements are read, it will be very proper for readers of all denominations, to make fome kind of eftimate of their improvement in the knowledge of magnitude, by taking a review of the whole. And this will be done with some astonishment by the judicious reader, when he confiders to what a hight he has raised himself above the vulgar in his conceptions upon this fubject; or indeed even above himself, when he compares the common perception which he fet out with, that Magnitudes which exactly agree together are equal, with the power of making a rectilineal figure equal to any given rectilineal figure, and fimilar to any other which may be given. But as an admiration of the great abilities of the author, and of his judicious arrangement will be the neceffary confequence of right notions upon this fubject; I fhall decline all encomiums upon these, to turn my thoughts to the affistance of those who may not have reafon to be fo well fatisfied with their improvement; which is likely to be most effectually given, by an enumeration of those circumstances, to which their want of fuccefs is most probably owing; an attention to which, upon fome future perufal, may remove every defect. And although I have mentioned them in the courfe of these differtations yet it may be useful as a conclufion to the whole to leave fuch an impreffion upon the reader's mind, as a fummary of the whole is likely to make. The reader therefore is to confider first of all whether he may not have had all this time, even a wrong notion of the very subject. The subject is magnitude, and not number; nor is it fufficient for him to be fatisfied that things equal to the fame are equal to one another; but he muft convince himself, not give his affent, that magnitudes which are equal to the fame, are equal to one another; and this is to be done by a particular examination, of as many different kinds of magnitudes magnitudes as he can eafily recollect, especially of fuch, as are thus compared in these first fix books. Next let him examine whether he has a proper notion of a definition, which he must do his endeavour to collect both from its nature and use; still keeping in his mind that every thing here refers to magnitude. And the fhewing that any property is confiftent with the definition is called a demonstration. But feveral demonstations ought to be carefully examined by a student in the circumftances of the person to whom I am now writing, entirely with a view to see whether he underftands what is meant by a geometrical demonftration; and as we suppose him to have his choice of the firft fix books; he ought especially to pick out the indirect demonftrations, which are of all others the most elegant and forcible, because they convince you contrary to the representation of the figure: the evidence of fuch propofitions therefore, if its force be perceived at all, comes to the mind purer, as it is not affected by the prejudices which the reafoning upon a particular figure, which feems to represent the confequences, is but to apt to introduce. For fuch reasonings are often disturbed even by turning the figure upfidedown; and I myself have often puzzled a student fufficiently acute, by such trials as joining AD in the twenty firft propofition of the first book, and requiring him to prove that AD and DC together are less than AB and BC together &c. The indirect demonftrations therefore are of all others the propereft for a student to exercise himself in; as this will be the means of removing that very great difficulty which learners find in making a ready and proper use of the suppofition; as his chief dependence for conviction in the indirect demonstrations, must be upon the fuppofition. It is true this may be got as I have remarked before, by an examination of the suppofition in every propofition, especially fuch as confist of several circumstances; and varying them, one by one, obferving the effect which every variation has upon demonstration; for by such a proceeding the force and ufe of the different parts of the fuppofition must be discovered, and indeed hardly by any other means. And in this progress he will be very much affifted by defcribing his figures will all the variety of pofitions, which his inftruments and the fuppofition will admit of, until he is certain that he has con quered |