B

F/E D

fary, and that he means by the inward angle of a figure, only the fum of two or more angles terminating at that angle; as, for example, the angle D of the quadrilateral figure ABCD, which is made Cup of the angles A D B, C D B. To this I anfwer, that the Jum of those two angles do not properly conftitute the angle D of the figure, but rather the fum of the angles A DH, CDH; becaufe [by def. 8. 1.] the angle at D, made by the right lines A D, C D, meeting in the print D, is the inclination [or opening] or leaning of the right lines A D, C D towards one another, and not their deviation or falling back or away from each other. Confequently, the angle at D of the quadrilateral figure ABCD, is, according to Euclid's definition of an angle, the angle ADC without the figure made by the inclination of its fides A D, CD; and fo are the angles B, E of the pentagon, or the angles D, E, G of the heptagon. And these must be taken to be fo, till we have a more general and diftinct definition of a right-lined angle. This being admitted, in any right-lined figure having both inward and outward angles, the fum of all the outward angles is greater than the fum of all the inward angles, by as many right angles as there are units in the product of the multiplication of the number of fides by the number of the outward angles, added to 4, and leffened by twice the number of fides of the figure; as, if the figure has four fides, it can have one outward angle. And in this cafe, the outward angle is equal to all the three inward angles. If a pentagon has two outward angles and three inward ones, the fum of all the outward angles will be greater than the fum of all the inward angles, by four right angles; and fo of others.

If two triangles have two fides of the one equal to two fides of the other, each to each, and one angle of the one equal to one angle of the other, but not that angle contained under the equal fides; and if

the

the angles of each triangle oppofite to one of the equal fides be either both acute or both obtufe: I fay, the remaining fide of the cne triangle will be equal to the remaining fide of the other; and the remaining angles of the one equal to the remaining angles of the other, each to each.

Let A B C, D E F be the two triangles, having two fides A B, B C equal to two fides D E, EF, cach to each; and the angle B A C of

the one equal to

the angle EDF of

the other, not being the angles contained

under

the equal fides.

And if both the

other angles ACB,

DF E, oppofite to A

the equal fides

A B, DE, be either an acute or an obtufe angle: I fay, the remaining fides a C, D F will be equal; as alfo the remaining angles AC B, DFE will be equal; as alfo ABC. and D E F.

First, Let the equal angles B A C, E D F be acute, and the angles A CB, D F E be both obtuse.

For if the fides A C, DF of the triangles A B C, DEF be not equal, one of them must be the greater. Let the fide AC be greater than the fide DF; and continuing out the fide DF to H, take the fide D H equal to the fide A C, and join the right line EH.

Then because the fides A B, A C of the triangle ABC, are equal to the fides DE, DH of the triangle DEH, each to each; and the angles B A C, E D F contained under them are equal. Therefore [by 4. 1.] the remaining angles A CB, DHE, and A B C, DE H, of the triangles A B C, DEF, will be equal; as alfo the remaining fides B C, E H. Wherefore fince the fide B C [by fuppofition] is equal to the fide E F, the fide E F will be equal to the fide E H: therefore the angles EF H, EHF [by 5. 1.] will be equal to one another. But because [by fuppofition] the angle

ACB

ACB or DFE is obtufe; and fince [by 13. 1.] the angles DFE, EFH are equal to two right angles, the angle E FH will be less than one right angle, that is, will be acute: therefore the angle EHF, which has been proved to be equal to it, will be an acute angle too. But it is an obtufe angle by fuppofition, which cannot be. Therefore the fide AC of the triangle A B C, is not greater than the side DF of the triangle D E F.

After the fame manner we demonftrate, that the fide A C is not leffer than the fide D F; therefore the fides AC, DF are equal. In like manner alío we prove, that these fides are equal, when the equal angles B AC, EDF, of the triangle A B C, D E F, are both obtufe. Therefore becaufe these two triangles have three fides of the one equal to three fides of the other, the remaining angles of these two triangles will [by 4. 1.] be equal; that is, the angles a B C, DE F, and A CB, D F E.

Therefore if two triangles have two fides of the one equal to two fides of the other, each to each, and one angle of the one equal to one angle of the other, but not that angle contained under the equal fides; and if the angles of each triangle oppofite to one of the equal fides, be either both acute or both obtufe: I fay, the remaining fide of the one triangle will be equal to the remaining fide of the other; and the remaining angles of the one equal to the remaining angles of the other. Which was to be demonftrated.

PROP. V. THEO R.

If two triangles have two fides of the one equal to two fides of the other, each to each; and one angle of the one equal to one angle of the other, but not that angle contained under the equal fides; and if one of the angles oppofite to an equal fide in one of the triangles be obtufe, and the other angle in the other triangle oppofite to the correfpondent equal fide be acute; I fay, the fum of thefe obtufe and acute angles will be equal to two right angles; and the difference of the angles contained under the equal fides, will be equal to the difference of the faid acute and oblique angles.

Let

Let ABC, DEF be two triangles, having two fides

A B, B C of

B

E

the one e

qual to two fides D E, EF of the other, cach to each; and one angle BAC of

the one, equal to one A

angle EDF of the other; but not that contained under the equal fides: Alfo let one of the angles, as AC B, Oppofite to an equal fide A B, be obtufe, and the correfpondent angle DFE, oppofite to the other equal fide D E, be acute: I fay, the fum of the angles AC B, DFE will be equal to two right angles; and the difference of the angles A B C, DEF, contained by the equal fides A B, B C, and DE, EF, will be equal to the difference of the obtufe and acute angles ACE, DFE, oppofite to the equal fides A B, DE.

For about the angular point E, as a centre with the diftance D F, describe a circle cutting the fide DF of the triangle D E F in the point H, which will fall between the points D, F, because the angle DFE is fuppofed to be acute. Join E H.

Then because [by def. 15. 1.] the fide EH is equal to the fide EF, the angles EF H, EHF at the bafe, will [by 5. 1.] be equal. But fince [by fuppofition] the fide EF is equal to the fide B C; the fides HE, BC will be equal to one another: and fince A B alfo is equal to D E, and the angle B A C equal to the angle EDF [by fuppofition]; therefore the angle A CB will be [by 4. of this] equal to the angle DH E, as well as the fide A c equal to the fide DH. And becaufe [by 13. 1.] the angles DHE, EHE are equal to two right angles, and the angle D F E has been proved to be equal to the angle EHF, and the angle B CA to the angle DHE; therefore the angles A C B, DEF will be equal to two right angles.

Again, because all the angles of every triangle are [by 32. 1.] equal to two right angles; and the angle B A C

[by

[by fuppofition] is equal to the angle EDF; therefore will the fum of the angles ACB, A B C be equal to the fum of the angles DFE, DEF. Confequently, the difference of the angles DEF and A B C, will be equal to the difference of the angles ACB, EF D.

Therefore if two triangles have two fides of the one equal to two fides of the other, each to each; and one angle of the one equal to one angle of the other, but not that angle contained under the equal fides; and if one of the angles oppofite to an equal fide in one of the triangles be obtufe, and the other angle in the other triangle oppofite to the correfpondent equal fide be acute; I fay, the fum of these obtuse and acute angles will be equal to two right angles; and the difference of the angles contained under the equal fides, will be equal to the difference of the faid acute and oblique angles. Which was to be demonftrated.

PROP. VI. THEOR.

In every triangle, the angle contained under the perpendicular drawn from the angle oppofite to the bafe upon it, and the right line bifecting that angle, will be one half the difference of the angles at the bafe.

Let AFD be a triangle, whofe perpendicular, drawn

F

from the angle F

F

upon the bate, is

FC; and let the

right line F B bifect the angle A F D oppofite to

the bafe: I fay,

C the angle BF C contained under

the perpendicular F C and the right line F B, will be equal to one half the difference of the angles F A D, F D A at the bafe.

For, firft, let the perpendicular F c fall within the triangle. Then becaufe [by 32. 1.] the angles FA C, A FC, ACF are equal to two right angles, and the angle ACF [by fuppofition] is a right angle; the angles F AC, AFC