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Book VI. Produce EF, GF, to H, K, and join FA, FC: then be
cause the parallelograms ABCD, AEFG are similar, DA is to AB, as GA to AE: where
D a Cor. 19. fore the remainder DG is a to the 5.
remainder EB, as GA to AE: but
с b 32. 6. F; wherefore AF, FC are in the same straight line b.
Next, Let the parallelograms 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.
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 centre,” are left out, as the addition of some unskilful hand.
In the Greek, as also in the Latin translation, the words á stums, “ 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 águ 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 Meydan Evitats,
PROP. B. C. D. B. VI.
These three propositions are added, because they are frequentiy 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 soiid 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 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
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 shewn by many examples; the following is sufficient.
Let there be any plane rectilineal figure, as the triangle a 12. 11. ABC, 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; therefore b 4. 1. the base EB is equal b
the other angles equal to 4. 6. the other angles; there
F 1.def. fore these triangles are 6. similard; in the same manner the triangle EBC is similar to
the triangle FBC and the triangle EAC to FAC; therefore Book XI. 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 mani. fest, 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 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
Book XI. 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 togtther 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 required that B, C, D together be greater than A, from each of these taking away B, C, the remaining one D must be greater than the excess of A above B and C: take therefore any massa nitude D which is less than A, B, C together, but greater than the excess of A above B and C: than B, C, D together are greater than A ; and because A is greater than either B or C much more will A and D, together with either of the two B, C be greater than the other: and, by the construction, A, B, C are together greater than D.
Cor. Il besides it be required, that A and B together shall not be less than C and D together; the excess of A and B together above C must not be less than D, that is, D must not be greater than that excess.
PROP. II. PROBLEM.
Four magnitudes A, B, C, D being given, of which A and B together are not less than C and Ď together, and such that any three of them whatever are greater than the fourth ; it is required to find a fifth magnitude E such, that any two of the three A, B, E shall be greater than the third, and also that any two of the three C, D, E shall be greater than the third. Let A be not less than B: and C not less than D.
First, Let the excess of C above D be not less than the excess of A above B : it is plain that a maguitude E can be taken which is less than the sum of C and D, but greater than the excess of C above D; let it be taken : then E is
likewise than the excess of A above B; wherefore F and B together are greater than A; and A is not less than B : therefore A and E together are greater than B: and, by the hypothesis, A and B together are not less than C and D together, and C and D together are greater than E; therefore likewise A and B are greater than E.