N in the given ftraight line AB produced, fo as to make the Book VI. rectangle AN, NB equal to a given fpace: Or, which is the in fame thing, having given AB the difference of the fides of a rectangle, and the magnitude of it, to find the fides. PROP. XXXI. B. VI. In the demonstration of this, the inverfion of proportionals is twice neglected, and is now added, that the conclufion may be legitimately made by help of the 24th prop. of b. 5. as Clavius had done. PROP. XXXII. B. VI. The enunciation of the preceding 26th prop. is not general enough; because not only two fimilar parallelograms that have an angle common to both, are about the fame diameter; but likewife two fimilar parallelograms that have vertically oppofite angles, have their diameters in the fame ftraight line: But there feems to have been another, and that a direct demonstration of thefe cafes, to which this 32d propofition was needful: And the 32d may be otherwife and fomething more briefly demonstrated as follows. PROP. XXXII. B. VI. If two triangles which have two fides of the one, &c. GF, proportional to the two fides FH, HC, viz. AG to GF, as alternate angles AGF, FKC are c K b 30. I. C qual: And AG is to GF, as (FH to HC, that is ) CK to KF; c 34. 1. wherefore the triangles AGF, CKF are equiangular, and the a 6. 6. angie AFG equal to the angle CFK: But GFK is a straight line, therefore AF and FC are in a ftraight line. The 26th prop. is demonftrated from the 32d, as follows. If two fimilar and fimilarly placed parallelograms have an angle common to both, or vertically oppofite angles; their diameters are in the fame ftraight line. Y 2 Firf € 14. I. Book VI. First, Let the parallelograms ABCD, AEFG have the angle BAD common to both, and be fimilar, and fimilarly placed; ABCD, AEFG are about the fame diameter. Produce EF, GF, to H, K, and join FA, FC: Then because the parallelograms ABCD, AEFG are fimilar, DA is to AB, as GA to AE; where a Cor. 19. fore the remainder DG is to the A 5. G D remainder EB, as GA to AE: But F H and AE to GF: Therefore as FH b 32. 6. B to HC, fo is AG to GF; and K Next, Let the parallelograms KFHC, GFEA, which are fimilar and fimilarly placed, have their angles KFH, GFE vertically oppofite; their diameters AF, FC are in the fame ftraight line. Because AG, GF are parallel to FH, HC, and that AG is to GF, as FH to HC; therefore AF, FC are in the fame ftraight line. PROP. XXXIII. B. VI. The words "because they are at the centre," are left out, as the addition of fome unfkilful hand. In the Greek, as alfo in the Latin tranflation, the words Η ετυχε, any whatever," are left out in the demonftration of both parts of the propofition, and are now added as quite neceffary; and, in the demonstration of the fecond part, where the triangle BGC is proved to be equal to CGK, the illative particle apa in the Greek text ought to be omitted. The fecond part of the propofition is an addition of Theon's, as he tells us in his commentary on Ptolomy's Μεγάλη Συντάξες, P. 50. PROP. B. C. D. B. VI. Thefe three propofitions are added, because they are frequently made ufe of by geometers, DEF. Book XI. THE DE F. IX. and XI. B. XI. HE fimilitude of plane figures is defined from the equality of of their angles, and the proportionality of the fides about the equal angles; for from the proportionality of the fides only, or only from the equality of the angles, the ' fimilitude of the figures does not follow, except in the cafe when the figures are triangles: The fimilar pofition of the fides, which contain the figures, to one another, depending partly upon each of thefe: And, for the fame reason, thofe are fimilar folid figures which have all their folid angles equal, each to each, and are contained by the fame number of fimilar plane figures: For there are fome folid figures contained by fimilar plane figures, of the fame number, and even of the fame magnitude, that are neither fimilar nor equal, as fhall be demonstrated after the notes on the 10th definition: Upon this account it was neceffary to amend the definition of fimilar folid figures, and to place the definition of a folid angle before it: And from this and the 10th definition, it is fufficiently plain how much the elements have been spoiled by unkilful editors. DE F. X. B. XI. Since the meaning of the word "equal" is known and established before it comes to be used in this definition; therefore the propofition which is the 10th definition of this book, is a theorem, the truth or falfehood of which ought to be demonftrated, not affumed; fo that Theon, or fome other Editor, has ignorantly turned a theorem which ought to be demonftrated into this roth definition: That figures are fimilar, ought to be proved from the definition of fimilar figures; that they are equal ought to be demonftrated from the axiom," Magnitudes that wholly coincide, are equal "to one another;" or from prop. A. of book 5. or the 9th prop. or the 14th of the fame book, from one of which the equality of all kind of figures must ultimately be deduced. In the preceding books, Euclid has given no definition of equal figures, and it is certain he did not give this: For what is called the ift def. of the 3d book, is really a theorem in which these circles are faid to be equal, that have the straight lines from their centres to the circumferences equal, which is plain, from the definition of a circle; and therefore has by 340 Book VI. the definitions. The First, Let the parallelograms ABC ned, but demonftrated: ~ BAD common to both, and be ffolid figures contained by ABCD, AEFG are about the f aal plane figures are equal to Produce EF, GF, to H, My deferve to be blamed who cause the parallelograms a Cor. 19, fore the remainder DG 5. b 32.6. 68.2. 54.6. I. propofition which ought to be is to AB, as GA to Aopofition be not true, muft it not have, for thefe thirteen hundred elementary matter? And this should acknowledge how little, through the we are able to prevent miftakes even in ences which are justly reckoned amongst the that the propofition is not universally true, remainder EB, as GA, FH, HC are r by many examples: The following is fufficient. any plane rectilineal figure, as the triangle Dght angles to the plane ABC; in DE take DE, DF Next lar ar opp G be equal to one angles FA, another, upon the oppofite fides of the plane, any point in EF; join DA, DB, DC; EA, FB, FC; GA, GB, GC: Because the ftraight with DA, DB, DC which it meets in that plane; and Je EDF is at right angles to the plane ABC, it makes right in the triangles EDB, FDB, ED and DB are equal to FD and each to each, and they contain right angles; therefore the bafe EB is equal b 10 DB, the bafe FB; in the C 1. def, fore thefe triangles are 6. fimilar: In the fame manner the triangle EBC is fimilar to the NOTES. 'IX 343 NOTE S. and the triangle EAC to PAC; therefore Book XI. , and their bafes the ftraight lines AB, COR. From this it appears that two unequal folid angles may be contained by the fame number of equal plane angles. For the folid angle at B, which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the folid angle at the fame point B which is contained by the four plane angles FBA, FBC, GBA, GBC; for this laft contains the other: And each of them is contained by four plane angles, which are equal to one another, each to each, or are the felf fame; as has been proved: And indeed there may be innumerable folid angles all unequal to one another, which are each of them contained by plane angles that are equal to one another, each to each: It is likewife manifeft that the before-mentioned folids are not fimilar, fince their folid angles are not all equal. And that there may be innumerable folid angles ali unequal to one another, which are each of them contained by the fame plane angles difpofed in the fame order, will be plain from the three following propofitions. Three magnitudes, A, B, C being given, to find a fourth fuch, that every three fhall be greater than the remaining one. Let D be the fourth; therefore D must be less than A, B, Y 4 B2 |