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because the straight lines FK, HG are at right angles to FH, and KG at right angles to A F K B
M FK; KG is equal to HF, by cor. pr. 2. that is, to N the double of EF. In the same manner, if AG be
TH produced to L, so that D +. GL be equal to AG, and pi LM be drawn perpendicular to AB, then LM is double of GK, and so on. In AD take AN equal to FE, and AO equal to KG, that is, to the double of FE, or AN; also take AP equal to LM, that is to the double of KG, or AO; and, let this be done till the straight line taken be greater than AD; let this straight line so taken be AP, and because AP is equal to LM, therefore LM is greater than AD. Which was to be done.
If two straight lines AB, CD make equal angles EAB, ECD with another straight line EAC towards the same parts of it; AB and CD are at right angles to some straight line.
Bisect AC in F, and draw FG perpendicular to AB: take CH in the straight line CD equal to AG, and on the contrary side of AC to that on which AG is, and join FH: therefore in the triangles AFG, CFH, the sides FA, AG are equal to FC, CH, each to each, and the angle FAG, that (15. 1.) is EAB is equal to the angle FCH; wherefore (4. 1.) the angle AGF is equal to CHF, and AFG to the angle CFH: to these last add the common angles AFH; therefore the two an- gles AFG, AFH are equal to the two angles CFH, HFA, which two last are equal together to two right angles (13. 1.), therefore also AFG, AFH are equal to two right angles, and consequently (14. 1.) GF and c H. FH are in one straight line. And because AGF is a right angle, CHF which is equal to it is also a right angle; therefore the straight lines AB, CD are at right angles to GH.
If two straight lines AB, CD, be cut by a third ACE so as to make the interior angles BAC, ACD, on the same side of it, together less than two right angles; AB and CD being produced shall meet one another towards the parts on which are the two angles which are less than two right angles.
At the point Ċ in the straight line CE make (23. 1.) the angle ECF equal to the angle EAB, and draw to AB the straight line CG at right angles to CF: then, because the angles ECF, EAB
are equal to one another, and
B FCA is greater than ACD and CD falls between CF and AB: and because CF and CD make an angle with one another, by prop. 3, a point may be found in either side of them CD, from which the perpendicular drawn to CF shall be greater than the straight line CG. Let this point be H, and draw HK perpendicular to CF, meeting AB in L: and because AB, CF contain equal angles with AC on the same side of it, by prop. 4, AB and CF are at right angles to the straight line MNO, which bisects AC in N and is perpendicular to CF; therefore, by cor. prop. 2, CG and KL which are at right angles to CF are equal to one another; and HK is greater than CG, and therefore is greater than KL, and consequently the point H is in KL produced. Wherefore the straight line CDH drawn betwixt the points C, H, which are on contrary sides of AL, must necessarily cut the straight line AL.
PROP. XXXV. B. I.
The demonstration of this proposition is changed, because, if the method which is used in it was followed, there would be three cases to be separately demonstrated, as is done in the translation from the Arabic; for, in the Elements, no case of a proposition that requires a different demonstration, ought to be omitted. On this account we have chosen the method which Mons. Clairault has given, the first of any, as far as I know, in his Elements, page 21, and which afterwards Mr. Simson gives in his page 32. But whereas Mr. Simson makes use of prop. 26, b. 1, from which the equality of the two triangles does not immediately follow, because, to prove that, the 4th of b. 1, must likewise be made use of, as may be seen in the very same case in the 34th prop. b. 1, it was thought better to make use only of the 4th of b. 1.
PROP. XLV. B. I.
The straight line KM is proved to be parallel to FL from the 33d prop.: whereas KH is parallel to FG by construction, and KHM. FGL, have been demonstrated to be straight lines. A corollary is added from Commandine, as being often used.
PROP. XIII. B. II. In this proposition only acute angled triangles are mentioned, whereas it holds true of every triangle ; and the demonstrations of the cases omitted are added : Commandine and Clavius have like. wise given their demonstrations of these cases.
PROP. XIV. B. II.
In the demonstration of this, some Greek editor has ignorantly inserted the words “but if not, one of the two BE, ED is the greater ; let BE be the greater, and produce it to F,” as if it was of any consequence whether the greater or lesser be produced: therefore, instead of these words, there ought to be read only, “but if not, produce BE to F."
PROP. I. B. III.
Several authors, especially among the modern mathematicians and logicians, inveigh too severely against indirect or apagogic demonstrations, and sometimes ignorantly enough, not being aware that there are some things that cannot be demonstrated any other way: of this the present proposition is a very clear instance, as no direct demonstration can be given of it: because, besides the definition of a circle, there is no principle or property relating to a circle antecedent to this problem, from which either a direct or indirect demonstration can be deduced : wherefore it is necessary that the point found by the construction of the problem be proved to be the centre of the circle, by the help of this definition, and some of the preceding propositions : and because, in the demonstration, this proposition must be brought in, viz. straight lines from the centre of a circle to the circumference are equal, and that the point found by the construction cannot be assumed as the centre, for this is the thing to be demonstrated; it is manifest some other point must be assumed as the centre; and if from this assumption an absurdity follows, as Euclid demonstrates there must, then it is not true that the point assumed is the centre; and as any point whatever was assumed, it follows that no point, except that found by the construction, can be the centre, from which the necessity of an indirect demonstration in in this case is evident.
PROP. XIII. B. III.
As it is much easier to imagine that two circles may touch one another within, in more points than one, upon the same side, than upon opposite sides; the figure of that case ought not to have been omitted ; but the construction in the Greek text would not have suited with this figure so well, because the centres of the circles must have been placed near to the circumferences ; on which account another construction and demonstration is given, which is the same with the second part of that which Campanus has translated from the Arabic, where, without any reason, the demonstration is divided into two parts.
PROP. XV. B. III. The converse of the second part of this proposition is wanting, though in the preceding, the converse is added, in a like case, both in the enunciation and demonstration; and it is now added in this. Besides, in the demonstration of the first part of this
15th, the diameter AD (see Commandine's figure) is proved to be greater than the straight line BC, by means of another straight Jine MN; whereas it may be better done without it: on which accounts we have given a different demonstration, like to that which Euclid gives in the preceding 14th, and to that which Theodosius gives in prop. 6, b. 1, of his Spherics in this very affair.
PROP. XVI. B. III. In this we have not followed the Greek nor the Latin translation literally, but have given what is plainly the meaning of this proposition, without mentioning the angle of the semicircle, or that which some call the cornicular angle, which they conceive to be made by the circumference and the straight line which is at right angles to the diameter, at its extremity; which angles have furnished matter of great debate between some of the modern geometers, and given occasion of deducing strange consequences from them, which are quite avoided by the manner in which we have expressed the proposition. And in like manner, we have given the true meaning of prop. 31, b. 3, without mentioning the angles of the greater or lesser segments: these passages Vieta, with good reason, suspects to be adulterated, in the 386th page of his Oper. Math.
PROP. XX. B. III. The first words of the second part of this demonstration, • xexhastw on nahi," are wrong translated by Mr. Briggs and Dr. Gregory “Rursus inclinetur;" for the translation ought to be “Rursus inflectatur;" as Commandine has it: a straight line is said to be inflected either to a straight or curve line, when a straight line is drawn to this line from a point, and from the point in which it meets it, a straight line making an angle with the former is drawn to another point, as is evident from the 90th prop. of Euclid's Data : for this the whole line betwixt the first and last points, is inflected or broken at the point of inflection, where the two straight lines meet. And in the like sense two straight lines are said to be inflected from two points to a third point, when they make an angle at this point; as may be seen in the description given by Pappus Alexandrinus of Apollonius's books de Locis planis, in the preface to his 7th book: we have made the expression fuller from the 90th prop. of the Data.
PROP. XXI. B. III. There are two cases of this proposition, the second of which, viz, when the angles are in a segment not greater than a semicircle, is wanting in the Greek: and of this a more simple demonstration is given than that which is in Commandine, as being derived only from the first case, without the help of triangles.
PROP. XXIII. and XXIV. B. III. In proposition 24 it is demonstrated, that the segment AEB must coincide with the segment CFD, (see Commandine's figure) and that it cannot fall otherwise, as CGD, so as to cut the other circle in a third point G, from this, that, if it did, a circle could cut another in more points than two: but this ought to have been proved to be impossible in the 23d prop. as well as that one of the segments cannot fall within the other: this part then in left out in the 24th, and put in its proper place, the 23d proposition.
PROP. XXV. B. III. This proposition is divided into three cases, of which two have the same construction and demonstration; therefore it is now divided only into two cases.
PROP. XXXIII. B. III. This also in the Greek is divided into three cases, of which two, viz. one in which the given angle is acute, and the other in which it is obtuse, have exactly the same construction and demonstration; on which account, the demonstration of the last case is left out as quite superfluous, and the addition of some unskilful editor; besides, the demonstration of the case when the angle given is a right angle, is done a round about way, and is therefore changed to a more simple one, as was done by Clavius.
PROP. XXXV. B. III. As the 25th and 33d propositions are divided into more cases, so this thirty-fifth is divided into fewer cases than are necessary. Nor can it be supposed that Euclid omitted them because they are easy; as he has given the case, which by far is the easiest of them all, viz. that in which both the straight lines pass through the centre; and in the following proposition he separately demonstrates the case in which the straight line passes through the centre, and that in which it does not pass through the centre: so that it seems Theon, or some other, has thought them too long to insert: but cases that require different demonstrations, should not be left out in the Elements, as was before taken notice of: these cases are in the translation from the Arabic, and are now put into the text.
PROP. XXXVII. B. III. At the end of this the words “in the same manner it may be demonstrated, if the centre be in AC," are left out as the addition of some ignorant editor.
DEFINITIONS OF BOOK IV. When a point is in a straight line, or any other line, this point is by the Greek geometers said awted Jav, to be upon, or in that line, and when a straight line or circle meets a circle any way, the one is said ATTEOSA, to meet the other: but when a straight Jine or circle meets a circle so as not to cut it, it is said sparsosa, to touch the circle: and these two terms are never promiscuously used by them: therefore in the fifth definition of book 4, the compound spaTTN50 must be read, instead of the simple artiTA); and