Book I. ftraight line CFD is nearer to the straight line AB at the point F

2. 4. I.

AB, and then goes further from it, before it cuts it, which is impoffible, and the fame thing will follow, if FE be faid to be greater than CA, or DB. therefore FE is not unequal to AC, that is, it is equal to it.

PROP. 2.

If two equal straight lines AC, BD be each at right angles to the same straight line AB; the straight line CD which joins their extremities makes right angles with AC and BD.

Join AD, BC; and because in the triangles CAB, DBA, CA, AB are equal to DB, BA, and the angle CAB equal to the angle DBA; the bafe BC is equal to the bafe AD. and in the triangles ACD, BDC, AC, CD are equal to BD, DC, and the base AD is equal to the bafe BC, therefore the b. 8. 1. angle ACD is equal to the angle C BDC. from any point E in AB draw

EF unto CD, at right angles to AB;
therefore, by Prop. 1. EF is equal to
AC, or BD; wherefore, as has been A

juft now fhewn, the angle ACF is e

F Ꭰ

E

B

qual to the angle EFC. in the fame manner the angle BDF is equal to the angle EFD; but the angles ACD, BDC are equal, there c.10. Def.1.fore the angles EFC and EFD are equal, and right angles; wherefore alfo the angles ACD, BDC are right angles.

COR. Hence, if two straight lines AB, CD be at right angles to the fame straight line AC, and if betwixt them a straight line BD be drawn at right angles to either of them, as to AB; then BD is equal to AC, and BDC is a right angle.

If AC be not equal to BD, take BG equal to AC, and join CG. therefore, by this Propofition, the angle ACG is a right angle; but ACD is also a right angle, wherefore the angles ACD, ACG are

equal

equal to one another, which is impoffible. therefore BD is equal to Book I. AC; and by this Propofition BDC is a right angle.

PROP. 3.

If two ftraight lines which contain an angle be produced, there may be found in either of them a point from which the perpendicular drawn to the other fhall be greater than any given straight line.

Let AB, AC be two straight lines which make an angle with one another, and let AD be the given ftraight line; a point may be found either in AB or AC, as in AC, from which the perpendicular drawn to the other AB fhall be greater than AD.

In AC take any point E, and draw EF perpendicular to AB; produce AE to G fo that EG be equal to AE; and produce FE to H, and make EH equal to FE, and join HG. because, in the tri

angles AEF, GEH, AE, EF are equal to GE, EH, each to each,

a

and contain equal angles, the angle GHE is therefore equal to a. 15. 1. the angle AFE which is a right angle. draw GK perpendicular to b.4. 1.

AB; and because

to the double of FE. in the fame manner, if AG be produced to L fo that GL be equal to AG, and 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; alfo 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.

PROP. 4.

If two ftraight lines AB, CD make equal angles EAB, ECD with another straight line EAC towards the fame parts of it; AB and CD are at right angles to fome straight line.

Bifect AC in F, and draw FG perpendicular to AB; take CH in the straight line CD equal to AG and on the contrary fide of AC to that on which AG is, and join FH. therefore, in the triangles AFG,

CFH

Book I. CFH the fides FA, AG are equal to FC, CH, each to each, and the angle FAG, that is EAB is equal

b. 4.

a

2. 15. 1. to the angle FCH; wherefore the angle AGF is equal to CHF, and AFG to the angle CFH. to these last add the common angle AFH, therefore the two angles AFG, AFH are equal

to the two angles CFH, HFA which

E

D

two last are equal together to two right CH

c. 13. 1. angles, therefore alfo AFG, AFH are

d. 14. 1. equal to two right angles, and confequently GF and FH are in one ftraight 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.

PROP. 5.

If two ftraight lines AB, CD be cut by a third ACE so as to make the interior angles BAC, ACD, on the fame fide of it, together lefs than two right angles; AB and CD being produced fhall meet one another towards the parts on which are the two angles which are lefs than two right angles.

a. 23. 1. At the point C in the ftraight line CE make the angle ECF equal to the angle EAB, and draw to AB the ftraight line CG at right angles to CF. then because the angles ECF, EAB are equal to one another, and that the

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 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.
cause AB, CF contain equal angles with AC on the fame fide of it,
by Prop. 4. AB and CF are at right angles to the straight line

MNO

MNO which bifects AC in N and is perpendicular to CF. therefore, Book I. 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 confequently the point H is in KL produced. Wherefore the straight line CDH drawn betwixt the points C, H which are on contrary fides of AL, muft neceffarily cut the straight line AB.

PROP. XXXV. B. I.

The Demonstration of this Propofition is changed, because if the method which is ufed in it was followed, there would be three cafes to be feparately demonftrated, as is done in the translation from the Arabic; for in the Elements no cafe of a Propofition that requires a different Demonstration ought to be omitted. On this account we have chofen the method which Monf. Clairault has given, the first of any, as far as I know, in his Elements, page 21. and which af terwards Mr. Simpfon gives in his, page 3 2. but whereas Mr. Simpfon makes ufe of Prop. 26. B. 1. from which the equality of the two triangles does not immediately follow, becaufe to prove that, the 4. of B. 1. must likewise be made ufe of, as may be feen, in the very fame cafe, in the 34. Prop. B. 1. it was thought better to make ufe only of the 4. of B. 1.

PROP. XLV. B. I.

The straight line KM is proved to be parallel to FL from the 33. Prop. whereas KH is parallel to FG by conftruction, and KHM, FGL have been demonftrated to be ftraight lines. a Corollary is added from Commandine, as being often used.

IN

PROP. XIII. B. II.

this Propofition only acute angled triangles are mentioned, whereas it holds true of every triangle. and the Demonftrations of the cafes omitted are added; Commandine and Clavius have likewife given their Demonstrations of these cafes.

PROP. XIV. B. II.

In the Demonftration of this, fome Greek Editor has ignorantly inferted the words, "but if not, one of the two BE, ED is the

Book II.

Book III. "greater; let BE be the greater and produce it to F," as if it was of any confequence 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."

Book III.

SEVER

PROP. I. B. III.

EVERAL Authors, especially among the modern Mathematicians and Logicians, inveigh too feverely against indirect, or A pagogic Demonftrations, and fometimes ignorantly enough; not being aware that there are fome things that cannot be demonftrated any other way. of this the prefent Propofition is a very clear instance, as no direct Demonstration can be given of it. because, befides 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 Demonftration can be deduced. wherefore it is neceffary that the point found by the conftruction of the Problem be proved to be the center of the circle, by the help of this Definition, and fome of the preceeding Propofitions, and because in the Demonstration, this Propofition must be brought in, viz. ftraight lines from the center of a circle to the circumference are equal, and that the point found by the construction cannot be affumed as the center, for this is the thing to be demonftrated; it is manifest some other point must be affumed as the center; and if from this affumption an abfurdity follows, as Euclid demonftrates there muft; then it is not true that the point affumed is the center; and as any point whatever was assumed, it follows that no point, except that found by the construction can be the center. from which the neceffity of an indirect Demonstration in this cafe 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 fame side, than upon oppofite fides; the figure of that cafe ought not to have been omitted; but the conftruction in the Greek text would not have futed with this figure fo well, because the centers of the circles must have been placed near to the circumferences. on which account another construction and demonstration is given which is the fame with the second part of that which Campanus has tranflated from

the