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 24. Prop. of B. 5. as Clavius had done. Book VI. PROP. XXXII. B. VI. The Enuntiation of the preceeding 26. Prop. is not general enough; because not only two fimilar parallelograms that have an angle common to both, are about the fame diameter; but likewise 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 Demonftration of thefe cafes, to which this 32. Propofition was needful. and the 32 may be otherwife and fomething more briefly demonftrated as follows. PROP. XXXII. B. VI. If two triangles which have two fides of the one, &c. Let GAF, HFC be two triangles which have two fides AG, GF proportional to the two fides FH, HC, viz. AG to GF, as FH to HC; and let AG be parallel to FH, and GF to HC; AF and FC are in a ftraight line. a A G K D H a. 31. 1. Draw CK parallel to FH, and let E it meet GF produced in K. because AG, KC are each of them parallel to FH, they are parallel b to one another, B and therefore the alternate angles AGF, FKC are equal. and AG is to GF, as (FH to HC, that is) CK to KF; wherefore the triangles AGF, CKF are equiangular 4, and the angle AFG equal to the angle CFK. but GFK is a straight line, therefore AF and FC are in a straight line ©. The 26. Prop. is demonftrated from the 32. 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. 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 b 30. I. c. 34. I. d. 6. 6. C. 14. I. Book VI. 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 a. Cor.19 5. AE; wherefore the remainder DG isa G to the remainder EB, as GA to AE. A D 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 straight 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 center," are left out, as the addition of fome unfkilful hand. In the Greek, as alfo in the Latin Tranflation, the words & TVXt," any whatever," are left out in the Demonftration of both parts of the Propofition, and are now added as quite necessary. 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 Meydan Zurtäğı, p. 50. PROP. B, C, D. B. VI. These three Propofitions are added, because they are frequently made ufe of by Geometers. Book XI. THER DEF. IX. and XI. B. XI. HE fimilitude of plane figures is defined from the equality of their angles, and the proportionality of the fides about the equal angles; for from the proportionality of the fides only, or only from from the equality of the angles, the fimilitude of the figures does Book XI. 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 these. and, for the fame reafon, 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 10. 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 10. Definition, it is fufficiently plain how much the Elements have been spoiled by unskilful Editors. DEF. X. B. XI. Since the meaning of the word " equal" is known and established before it comes to be ufed in this Definition, therefore the Propofition which is the 10. Definition of this Book, is a Theorem the truth or falfhood 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 10. 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 ano"ther;" or from Prop. A. of Book 5. or the 9. Prop. or the 14. of the fame Book, from one of which the equality of all kind of figures muft ultimately be deduced. In the preceeding Books, Euclid has given no Definition of equal figures, and it is certain he did not give this. for what is called the 1. Def. of the 3. Book, is really a Theorem in which thefe circles are faid to be equal, that have the ftraight lines from their centers to the circumferences equal, which is plain from the Definition of a circle, and therefore has by fome Editor been improperly placed among the Definitions. The equality of figures ought not to be defined, but demonftrated. therefore tho' it were true that folid figures contained by the fame number of fimilar and equal plane figures are equal to one another, yet he would justly deferve to be blamed who should make a Definition of this Propofition which ought to be demonftrated. But if this Propofition be not true, must it not be confeffed that Geome ters Book XI. ters have for these thirteen hundred years been mistaken in this Elementary matter? and this fhould teach us modefty, and to acknowledge how little, thro' the weakness of our minds, we are able to prevent mistakes even in the principles of fciences which are justly reckoned amongst the most certain; for that the Propofition is not univerfally true can be fhewn by many examples; the following is fufficient. a Let there be any plane rectilineal figure, as the triangle ABC, a. 12. 11. and from a point D within it draw the straight line DE at right angles to the plane ABC; in DE take DE, DF equal to one another, upon the opposite fides of the plane, and let G be any point in EF; join DA, DB, DC; EA, EB, EC; FA, FB, FC; GA, GB, GC. because the straight line EDF is at right angles to the plane ABC, it makes right angles with DA, DB, DC which it meets in that plane; and in the triangles EDB, FDB, ED and DB are equal to FD and DB, each to each, and they contain right angles, therefore the bafe EB is equal to the base FB; in the fame manner EA is equal to FA, and EC to FC. and in the triangles EBA, FBA, EB, BA are b. 4. I. equal to FB, BA, and the G bafe EA is equal to the E c. 8. 1. angle EBA is equal to the angle FBA, and the triangle EBA equal to the triangle FBA, and the other angles equal to the other angles; therefore thefe triangles are fimilard. in the fame manner the 4. 6. 1. Def. 6. B F triangle EBC is fimilar to the triangle FBC, and the triangle EAC to FAC. therefore there are two folid figures each of which is contained by fix triangles, one of them by three triangles the common vertex of which is the point G, and their bases the straight lines AB, BC, CA; and by three other triangles the common vertex of which is the point E, and their bafes the fame lines AB, BC, CA the other folid is contained by the fame three triangles the common vertex of which is G, and their bafes AB, BC, CA; and by three other triangles of which the common vertex is the point F, and their their bafes the fame ftraight lines AB, BC, CA. now the three Book XI. triangles GAB, GBC, GCA are common to both folids, and the three others EAB, EBC, ECA of the firft folid have been fhewn equal and fimilar to the three others FAB, FBC, FCA of the other folid, each to each. therefore thefe two folids are contained by the fame number of equal and fimilar planes. but that they are not equal is manifeft, because the firft of them is contained in the other. therefore it is not univerfally true that folids are equal which are contained by the fame number of equal and fimilar planes. 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 solid 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 cach. 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 all 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. PROP. I. PROBLEM. 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, C together. of the three A, B, C let A be that which is not lefs than either of the two B and C. and firft, let B and C together be not lefs than A; therefore B, C, D together are greater than A. and because A is not lefs than B; A, C, D together are greater than B. in the like manner A, B, D together are greater than C. wherefore in the cafe in which B and C together are not lefs than A, any magnitude D which is less than A, B, C together will answer the Problem. But if B and C together be lefs than A, then because it is re |