K C D F M L N B PO the square of EG; and consequently the straight line AK or GL is not greater than GE. E Now, if GE be equal to GL, the circle EHF touches AB in L, and therefore the square of AL is (36. 3.) equal to the rectangle EA, AH, that is, to the given rectangle C, D; and that which H was required is done: but if EG, GL be unequal, EG must A be the greater: and therefore the circle EHF cuts the straight line AB: let it cut it in the points M, N, and upon NB describe the square NBOP, and complete the rectangle ANPQ: because LM is equal to (3. 3.) LN, and it has been proved that AL is equal to LB; therefore AM is equal to NB, and the rectangle AN, NB equal to the rectangle NA, AM, that is, to the rectangle (Cor. 36. 3.) EA, AH, or the rectangle C, D: but the rectangle AN, NB is the rectangle AP, because PN is equal to NB: therefore the rectangle AP is equal to the rectangle C, D; and the rectangle AP equal to the given rectangle C, D has been applied to the given straight line AB, deficient by the square BP. Which was to be done. Q 4. To apply a rectangle to a given straight line that shall be equal to a given rectangle, exceeding by a square. Let AB be the given straight line, and the rectangle C, D the given rectangle, it is required to apply a rectangle to AB equal to CD, exceeding by a square. Draw AE, BF at right angles to AB, on the contrary sides of it, and make AE equal to C, and BF equal to D: join EF, and bisect it in G; and from the centre G, at the distance GE, describe a circle meeting AE again in H; join HF, and draw CL parallel to AE; let the circle meet AB produced in M, N, E is equal to LN, and AL to LB, therefore MA is equal to BN, and the rectangle AN, NB to MA, AN, that is (35. 3.) to the rectangle EA, AH, or the rectangle C, D: therefore the rectangle AN, NB, that is, AP, is equal to the rectangle C, D; and to the given straight line AB the rectangle AP has been applied equal to the given rectangle C, D, exceeding by the square BP. be done. Which was to Willebrordus Snellius was the first, as far as I know, who gave these constructions of the 3d and 4th problems, in his Apollonius Batavius; and afterwards the learned Dr. Halley gave them in the scholium of the 18th prop. of the 8th book of Apollonius's conics restored by him. The 3d problem is otherwise enunciated thus: to cut a given straight line AB in the point N, so as to make the rectangle AN, NB equal to a given space; or, which is the same thing, having given AB the sum of the sides of a rectangle, and the magnitude of it being likewise given, to find its sides. And the fourth problem is the same with this. To find the point N in the given straight line AB produced so as to make the rectangle AN, NB equal to a given space: or, which is the same thing, having given AB the difference of the sides of a rectangle, and the magnitude of it, to find the sides. PROP. XXXI. B. VI. In the demonstration of this, the inversion of proportionals is twice neglected, and is now added, that the conclusion 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 similar parallelograms that have an angle common to both, are about the same diameter; but likewise two similar parallelograms that have vertically opposite angles, have their diameters in the same straight line: but there seems to have been another, and that a direct demonstration of these cases, to which this 32d proposition was needful: and the 32d may be otherwise and something more briefly demonstrated as follows. PROP. XXXII. B. VI. If two triangles which have two sides of the one, &c. Let GAF, HFC be two triangles which have two sides AG, GF proportional to the two sides 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 straight line. Draw CK parallel (31. 1.) to FH, and let Α G D H it meet GF produced in K: because AG, E KC are each of them parallel to FH, they are parallel (30. 1.) to one another, and therefore the alternate angles AGF, FKC are equal and AG is to GF, as (FH to HC, that is 34. 1.) CL to KF; wherefore the triangles AGF, CKF are equiangular, (6. 6.) and the angle AFG equal to the angle CFK: but GFK is a straight line, therefore AF and FC are in a straight line (14. 1.). B K C The 26th prop. is demonstrated from the 32d, as follows: If two similar and similarly placed parallelograms have an angle common to both, or vertically opposite angles; their diameters are in the same straight line. First, let the parallelograms ABCD, AEFG have the angle BAD common to both, and be similar and similarly placed; ABCD, AEFG are about the same diameter. Α G D Produce EF, GF, to H, K, and join FA, FC: then because the parallelograms ABCD, AEFG are similar, DA is to AB, as GA to AE: wherefore the remainder DG is (Cor. 19. 5.) to the remainder EB, as GA to AE: but DG is equal to FH, EB to HC, and AE to GF: therefore as FH to HC, E so is AG to GF; and FH, HC are parallel to AG, GF; and the triangles AGF, FHC are joined at one angle in the point F: wherefore AF, FC are in the same straight line (32. 6.) B K C H Next, let the parallelograms KHFC, GFEA, which are similar and similarly placed, have their angles KFH, GFE vertically opposite; their diameters AF, FC are in the same 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 same straight line (32. 6.). PROP. XXXIII. B. VI. The words "because they are at the centre," are left out, as the addition of some unskilful hand. 66 In the Greek, as also in the Latin translation, the words a εTuxe 'any whatever," are left out in the demonstration of both parts of the proposition, and are now added as quite necessary; and in the demonstration of the second part, where the triangle BGC is proved to be equal to CGK, the illative particle aga in the Greek text ought to be omitted. The second part of the proposition is an addition of Theon's, as he tells us in his commentary on Ptolemy's Mayaλn Zuvrağıs, p. 50. PROP. B. C. D. B. VI. These three propositions are added, because they are frequently made use of by geometers. DEF. IX. and XI. B. XI. The similitude of plane figures is defined from the equality of their angles, and the proportionality of the sides about the equal angles; for from the proportionality of the sides only, or only from the equality of the angles, the similitude of the figures does not follow, except in the case when the figures are triangles: the similar position of the sides which contain the figures, to one another, depending partly upon each of these: and, for the same reason, those are similar solid figures which have all their solid angles equal, each to each, and are contained by the same number of similar plane figures: for there are some solid figures contained by similar plane figures, of the same number, and even of the same magnitude, that are neither similar nor equal, as shall be demonstrated after the notes on the 10th definition; upon this account it was necessary to amend the definition of similar solid figures, and to place the definition of a solid angle before it: and from this and the 10th definition, it is sufficiently 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 used in this definition; therefore the proposition which is the 10th definition of this book, is a theorem, the truth or falsehood of which ought to be demonstrated, not assumed; so that Theon, or some other editor, has ignorantly turned a theorem which ought to be demonstrated into this 10th definition; that figures are similar, ought to be proved from the definition of similar figures; that they are equal, ought to be demonstrated 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 same 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 1st def. of the 3d book is really a theorem in which these circles are said 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 some editor been improperly placed among the definitions. The equality of figures ought not to be defined, but demonstrated: therefore, though it were true, that solid figures contained by the same number of similar and equal plane figures are equal to one another, yet he would justly deserve to be blamed who would make a definition of this proposition, which ought to be demonstrated. But if this proposition be not true, must it not be confessed, that geometers have, for these thirteen hundred years, been mistaken in this elementary matter? And this should teach us modesty, and to acknowledge how little, through the weakness of our minds, we are able to prevent mistakes, even in the principles of sciences which are justly reckoned amongst the most certain; for that the proposition is not universally true, can be shown by many examples; the following is sufficient. Let there be any plane rectilineal figure, as the triangle ABC, and from a point within it draw (12. 11.) the straight line DE at right angles to the plane ABC; in DE take DE, DF equal to one another, upon the opposite sides 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 EBD, FBD, ED and DB are equal to FD and DB, each to each, and they contain right C angles; therefore the base EB is equal (4. 1.) to the base FB; in the same manner EA is equal to FA, and EC to FC: and in the triangles EBA, FBA, EB, BA are equal to FB, BA; and the base EA, is equal to the base FA; wherefore the angle EBA is equal (8. 1.) to the angle FBA, and the triangle EBA equal (4. 1.) to the triangle FBA, and the other angles equal to the other angles; therefore these triangles are similar (4. 6. 1. def.): in the same manner the triangle EBC is similar to the triangle FBC, and the triangle EAC to FAC; therefore there are two solid figures, each of which is contained by six 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 bases the same lines AB, BC, CA: the other solid is contained by the same three triangles the common vertex of which is G, and their bases AB, BC, CA; and by three other triangles of which the common vertex is the point F, and their bases the same straight lines AB, BC, CA: now the three triangles GAB, GBC, GCA are common to both solids, and the three others EAB, EBC, ECA of the first solid have been shown equal and similar to the three others FAB, FBC, FCA of the other solid, each to each; therefore these two solids are contained by the same number of equal and similar planes: but that they are not equal is manifest, because the first of them is contained in the other: therefore it is not universally true that solids are equal which are contained by the same number of equal and similar planes. COR. From this it appears that two unequal solid angles may be contained by the same number of equal plane angles. For the solid angle at B, which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the solid angle at the same point B, which is contained by the four plane angles FBA, FBC, GBA, GBC; for this last 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 self same; as has been proved: and indeed there may be innumerable solid 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 likewise manifest that the before mentioned solids are not similar, since their solid angles are not all equal. And that there may be innumerable solid angles all unequal to one another, which are each of them contained by the same plane angles disposed in the same order, will be plain from the three following in from the three propositions. |