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PROP. XXXI. B. VI.
In the demonstration of this, the inversion of proportionals is twice neglected, and is now adderl, 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 their 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. XXXır. B. VI.
If two triangles which have two sides of the one, &c.
a 31, 1. Draw CK parailel* to FH, and
E let it meet GF produced in K: because AG, KC are each of them
b 30. 1. parallel to FI, they are parallel! to one aiyother, and therefore the.
с alternate angles AGF, FKC are equal ; 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
d 6. 6. angle AFG equal to the angle CFK: but GFK is a straight line, therefore AF and FC are in a straight line
e 14. 1., The 26th prop. is demonstrated from the 32d, as follows:
If two similar and similarly placed parallelograms have an anogle common to both, or vertically opposite angles; their diameters are in the same straight line.
First, Let the parallelogran ABCD, AEFG have the angle BAD common to both, und be similar and similarly, placed; ABCD, AEFG are about the same diameter.
BOK VI. Produce EF, GF, to H, K, and join FA, FC:
then because the parallelograns ABCD, AEFG are similar, DA is to AB, ++
as GA to AE: wherefore the re. a Cor. 19.
EB, as GA to AE: but DG is equal
H therefore as FH to HC, 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; B wherefore AF, FC are in the same
K b 32. 6. straight line".
Next, Let the parallelograṁs KFHC, GFEA, which are similar and similarly placed, have their angles KFH, GFE vertically opposite; their diameters, AF, FC are in the same straight line.
1) 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.
PROP. XXXIII. B. VI. The words “ because they are at the centre,” are left out, as the addition of some-unskilful hand.
In the Greek, as also in the Latin translation, the words & stu?? 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 iş proved to be equal to CGK, the illative particle apt 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 Me, 1an Lurtas, 50,
PROP. B. C. D. B. VI.
These three propositions are added, because thay are frce. quently made use of by geometers.
DEF. IX. and XI. B. XI.
The similitude of plane figures is defined from the equally
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 'símilar 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.
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 kinds 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
Book XI. 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 e 12. 11. ABC, and from a point”, D within it drawe 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, FB, EC; FA, FB, FC; GA, GB, GC: because the straight line EDF is át 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 b 4. 1. the base EB is equal to the base FB ; in the
C:and in the triangles
wherefore the angle c 8. 1: EBA is equal to the
angle FBA, and the tri-
fore these triangles are 54.6. similard; in the same manner the triangle EBC is similar to d31. def.
the triangle FBC, and the triangle EAC to FAC; therefore Book XI.
Cor. From this it appears that two unequal solid angles may be contained by the same number of cqual plane angles.
Fof the solid angle at B, which is containied 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 propositions.
Three magnitudes, AçB, C being given, to find a fourth such, that every thrce shall be greater than the remaining one. "Let D be the fourth : therefore D must be less than A, B, C