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Book VI. 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 24. Prop. of B. 5. as Clavius had done.
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 same diameter; but likewise two similar parallelograms that have vertically cpposite angles, have their diameters in the fame straight line. but there seems to have been another, and that a direct Demonstration of these cases, to which this 3 2. Proposition was needful. and the 32 may be otherwise and something more briefly demonstrated as follows.
a. 31. I.
PROP. XXXII. B. VI. .
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,
b 30. 1.
К. and therefore the alternate angles
C AGF, FKC are equal. and AG is to GF, as (FH to HC, that is). c. 34. I. CK to KF; wherefore the triangles AGF, CKF are equiangular 4, d. 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
The 26. Prop. is demonstrated from the 32. as follows.
If two similar and similarly placed parallelograms have an angle common to both, or vertically opposite angles; their diameters are in the fame 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 fame diameter.
e. 14. I.
Book VI. Produce EF, GF, to H, K, and join FA, FC. then because the
w parallelograms ABCD, AEFG are similar, DA is to AB, as GA to a. Cor.19 5.AE; wherefore the remainder DG isa to the remainder EB, as GA to AE.
D but DG is equal to FH, EB to HC, and AE to GF. therefore as FH to E
H HC, so is AG to GF; ‘and FH, HC are parallel to AG, GF; and the triangles AGF, FHC are joined at one B
K angle, in the point F; wherefore AF, b. 32. 6. FC are in the same straight line b.
Next, Let the parallelograms KFHC, GFEA which are similar and fimilarly 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 b.
PROP. XXXIII. B. VI. The words“ because they are at the center,” are left out, as the addition of some unikilful hand.
In the Greek, as also in the Latin Translation, the words & TVXE," 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 äp 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 Prolomy's Μεγάλη Συνταξις, , p. 50.
PROP. B, C, D. B. VÌ. These three Propositions are added, because they are frequently made use of by Geometers.
DEF. IX. and XI.
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 Book XI. not follow, except in the case when the figures are triangles. them similar position of the sides, which contain the figures, to one another, depending partly upon each of these. and, for the same reafon, those are similar 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 some solid figures contained by similar plane figures; of the same number, and cven of the same magnitude, that are neither similar nor equal, as shall be demonstrated after the Notes on the 10. 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 1o. Definition, it is fufficiently plain how inuch the Elements have been spoiled by unskilful 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 10. Definition of this Book, is a Theorem the truth or falfhood of which onght to be demonstrated, not assumed ; so that Theon, or some other Editor, has ignorantly turned a Theorem which ought to be deinonstrated into this 10. Definition. that figures are fiinilar, ought to be proved from the Definition of limilar figures; that they are cqual ought to be demonstrated 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 must 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 these circles are said to be equal, that have the straight 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 demonstrated. therefore tho' it were true that solid figures contained by the fame number of fimilar and equal plane figures are equal to one another, yet he would justly deserve to be blamed who should make a Definition of this Proposition which ought to be demonstrated. But if this Proposition be got true, must it not be confessed that Geome
Book X1. ters have for these thirteen hundred years been mistaken in this
Elementary matter? and this should teach us modesty, and to acknowledge how little, thro' 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 shewn by many examples; the following is sufficient.
Let there be any plane rectilineal figure, as the triangle ABC, 2. 12. 11. and from a point D within it draw a 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 EDB, FDB, ED and DB are equal
to FD and DB, each to each, and they contain right angles, thereb. 4. 8. fore the base EB is equal b to the base FB; in the same manner EA is equal to FA, and EC to
angle FB A, and the tri-
their bafes the fame straight 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 first solid have been mewn equal and similar to the three others FAB, FBC, FCA of the other folid, each to each. therefore these two folids are contained by the fame 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 cqual 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 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 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 fame; 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 cach. it is likewise manifelt that the before mentioned solids are not limilar, since thcir solid 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 disposed in the same order, will be plain from the three fullowing Propositions.
PROP. I. PROBLEM, Three magnitudes A, B, C being given, to find a fourth such, that every three shall 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 less than either of the two B and C. and first, let B and C together be not less than A; therefore B, C, D together are greater than A. and because A is not less 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 case in which B and C together are not less than A, any magnitude D which is less than A, B, C together will answer the Problem. But if B and C together be less than A, then because it is re