Book XII. Because DC, GF are parallel, the angle AGC fhall be [by 29. 1.] equal to the angle AGF, and for the fame reafon because BC, EF are parallel, the angle ABC is equal to the angles A EF, and each of the angles BCD, E F G is equal to the oppofite angle DAB [by 34. 1.] and so are equal to one another; wherefore the parallelograms ABCD, AEFG are equiangular; and because the angle ABC is equal to the angle A EF, and the angle BAC is common to both the triangles BA C, E AF, they will be equiangular to one another: therefore [by 4. 6. Jas A B is to BC, fo is AE to EF; and because the oppofite fides of the parallelograms are [by 31. 1.] equal to one another, AB [by 9.5.] fhall be to A D as A E to AG, and DC to C B as G F to FE; and alfo CD to DA as FG to GA: therefore the fides of the parallelograms A B CD, A EFG about the equal angles are proportional, and they are therefore [by 1. def. 6.] fimilar to one another: for the fame reason the parallelograms A B C D, F HC K are fimilar; whefore the parallelograms G E, K H are each fimilar to the parallelogram D E. but right line figures that are fimilar to the fame right line figures, are fimilar to one another [by 21.6.]: therefore the parallelogram G E is fimilar to KH, which was to be demonftrated.

On PRO P. XXIX, XXX. BOOK VI.

It is faid in a late edition of the first fix, and eleventh and twelfth books of Euclid's Elements of Geometry, "That these two problems are the most general and use"ful of all in the elements, and are most frequently made "ufe of by the antient geometers in the folutions of other "problems." What use the antient geometricians might make of these two problems I cannot tell; but I think it is too much to fay, they are the moft general and ufeful of all in the elements; they are certainly lefs useful than many problems in the elements; and the problems of the antients, whatever they were, might be refolved without them.Their obfcure wording makes them neither palatable nor eafily retainable in the memory. In short, they are difagreeable. and not very useful.

On PROP. XVIII. of the Additions to BOOK VI.

Towards the latter end of the demonftration of this propofition, there is fomething omitted which should not, to make the demonftration fufficiently clear. The inference; therefore the ratio of the angle EDV to the angle EDA, will be greater than the ratio of GE to E A, is not made out from what has been already said in the demonftration fo as to be readily apprehended by the reader.-Now, in order to supply the intermediate fteps that have been omitted, we must begin with affuming what has been clear enough demonftrated [by prop. 8. book 5.], viz. That the ratio of the fector EDT, to the fector EDH, will be greater than that of the triangle TZD to the triangle EDA. This being granted,the ratio of twice the sector EDT, to the fector E D H, will be greater than the ratio of twice the triangle E Z D, to the triangle EDA; and adding the equal ratios of the sector H DE to the fector H DE, and that of the triangle A D E, to the triangle ADE, to each of these unequal ratios, the compound ratio of twice the fector HD E, together with the sector ED T to the sector H D E, will be greater than the ratio of twice the triangle E DZ, together with the triangle A DE, to the triangle ADE: but the sector E D V is equal to twice the fector EDT added to HDE, and the line E G equal to twice the line Ez added to A E: wherefore the ratio of the sector DEV to the sector H E, will be greater than the ratio of the triangle EDG to the triangle A DE.

Note, That the fector E D V is equal to the fector A DE, together with twice the sector EDV follows, by drawing a right line DQ to the point Q, where AG cuts the arch HEV; for then it eafily appears that the fector T D Q is equal to the sector EDT, and the sector Q D v equal to the fector HDE: alfo the triangle zDQ is equal to E DZ, and the triangle QDG equal to the triangle A D E. I think no more need be added to make the demonftration fufficient. ly clear.

On D E F. VIII. BOOK XI.

Some word this definition thus; "parallel planes are "fuch which do not meet one another, tho' produced.”--

*See Dr. Robert Simplon's Euclid,

But the words, Infinitely, or Never fo far produced, should have been added, otherwife the definition will not be good: for two planes which are not parallel may be produced, and not meet, for want of being produced far enough.

The 35th definition of the first book of parallel straight lines has the words produced never fo far,' in it; and fo fhould this 8th definition of parallel planes have these words in it.

On DE F. IX, X. BOOK XI.

The ninth definition wants amending. It should be fi milar folid figures, are those which are contained under equal numbers of fimilar plane figures, having fimilar inclinations each to each. For it is not always true, without the additional condition of the planes being fimilarly inclined. But when any one of the folid angles of the two fimilar folids confifts of not more than three plane angles, the definition will want no amending.- Euclid himself might have given the ninth definition without the reftriction above-mentioned: not through ignorance, but because he knew it to be true and fufficient for his purpose, in the demonstrations of the 11th and 12th books, where the folid angles of the folid prifms and parallelopipedons do not confift of more than three plane angles. Thefe elements contain only the theory of the most useful and fimple folids (two of the five Platonick bodies perhaps being excepted) and not of ufelefs polyhedrons, and irregular folids, whose solid angles are made up of many plane angles more than three; and for this reafon Euclid might confine and adapt his ninth definition only to the eftablishment of the elementary propofitions of the most fimple folids-he thought the shortest definition was beft, although neither general nor exact, when there was no occafion for a longer. He has done fomething like this in the 2d definition to the 3d book, where he fays, "A ftraight line is faid to touch a circle when it "meets the circle, and being produced, does not cut it."

Now this definition of a tangent line is only true with regard to a circle and the three conic fections; the parabola, ellipfis and hyperbola, being all curve lines of the firft

and

and moft fimple order. But with respect to the curve lines of the fecond, and higher orders, their tangents will not be included within this Euclidean definition of a tangent-for a right line will both touch one of these curves of the fecond order, and cut it at the fame time, if it be produced. Therefore Euclid's definition of a tangent to a circle is only particular and aright. But generally with regard to other curve lines it may be wrong and defective. It may be Euclid himself might have known this, but did not care to trouble himself with a general and more complex definition of a tangent to the higher orders of curve lines, of no ufe in his elements, he only treating of circles and their tangents. Moreover, Euclid has given the definition only of right cones and cylinders, and not of all cones in general, because he only treats of right cones and cylinders in his 12th book.

There are fome [as Dr. Robert Simpfon in his Euclid] who otherwife define folid fimilar figures in general thus:"They are fuch that have all their folid angles equal each` "to each, and which are contained by the fame number "of fimilar planes."

Now this definition appears to me to be true indeed, but more obfcure than that above, because of the words, equal folid angles, which have not yet been defined. To remedy which in part, Dr. Simpfon has put Euclid's 11th definition of a folid angle into the place of his 9th, and put this definition of his into the place of Euclid's 11th. But what are equal folid angles the Doctor does not exprefsly tell us: and no definition can be clear, if all the words in it be not foreknown and well understood.

Thus much of the 9th definition. Next to the 10th.This is for ever true, when all the folid angles of the two folids confift of no more than three plane angles; but when the folid angles have more than three plane angles in order for the definition to be true, it must be thus expreffed, viz. Equal and fimilar folid figures are those which are contained by fimilar planes of the jame number and magnitude fimilarly inclined each to each.

But, as I have obferved already, Euclid had no occafion for this general definition, because his is fufficient for all the fimple folids that he treats of, and their demonstrations, in thefe his elements; and no man can be juftly

Gg 2

blamed

Book V. blamed for omitting lefs evident and almost useless definitions for his purpose, whether true or falfe, and using another more fimple and eafy, though not true, if it be applied to the further extenfion of the theory of irregular polyhedrons, which Euclid had not a mind to do. Let those who have wrote more upon the fubject give other definitions fuiting their propofitions; Euclid thought this fufficient for his.

Who

Solid geometry from its nature is difficult, by the great number of lines, and planes of different inclinations, that are to be confidered in it: and hence it has happened that there are but a very few who have cared to meddle with it, and carry their fpeculations herein farther than Euclid has done, the fpheric and conic fection-writers excepted. And many have but flightly looked over Euclid's 11th and 12th books. They have taken all for true, without any strict examination of the definitions and demonftrations, ever does not perceive the usefulness of a subject before he enters into it will never much confider it. And the great Newton himself, in his univerfal arithmetic, fays, geometrical propofitions are just as valuable as ufeful. If therefore Euclid, or any body elfe, had not known the falfhood of the ninth and tenth definitions, when extended to irregular polyhedrons, which no man cared to confider, by reafon of the difficulty and apparent unusefulness of such fpeculations, they could not be blamed in a greater degree than what must arife from the importance of the subject. But as this is fmall, fo muft the errors. Pursuits of little value, either from ignorance or defign, are infignificant.

Dr. Simpson fays, "The tenth definition of the eleventh "book of Euclid is a theorem that should be demonftrat

ed, and not a definition; and that Theon, or fome "other editor, ignorantly put it down amongst the defi"nitions."---Now, granting it to be a theorem, yet it is one of fo evident a nature, from the contemplation of two equal folids, that totally agree when they are put into one another as to require no demonftration. In this case, who does not evidently perceive that the folids being equal and agreeing, their boundaries, viz. the planes, and their inclinations or folid angles, muft agree too, that is, be equal and fimilar each to each.