straight line makes the adjacent angles equal to one another, each of the angles is called a RIGHT ANGLE and the straight line which stands on the other is called a PERPENDICULAR. Fig. 4. 8. An OBTUSE ANGLE is that which is greater than a right angle. Fig. 5. 9. An ACUTE ANGLE is that which is less than a right angle. Fig. 6. 10. PARALLEL STRAIGHT LINES are fuch as are in the same plane, and which being produced ever so far both ways do not meet. Fig. 7. 11. A FIGURE is that which is inclosed by one or more boundaries. 12. RECTILINEAL FIGURES are those which are contained by ftraight lines. 13. Every plane figure bounded by three straight Jines is called a TRIANGLE, of which the three straight lines are called the fides, that fide upon which the triangle is conceived to stand is called the base, and the opposite angular point the ver tex. 14. An EQUILATERAL TRIANGLE is that which has three equal fides. Fig. 8. 15 An ISOSCELES TRIANGLE is that which has only two equal fides. Fig 9. 16 A SCALENE TRIANGLE is that which has all its fides unequal. Fig. 11. 17. A RIGHT ANGLED TRIANGLE is that which has a right angle. Fig. 10. 18. An OBTUSE ANGLED TRIANGLE is that which has an obtuse angle. Fig. 11. 19. An ACUTE ANGLED TRIANGLE is that which has all its angles acure. Fig. 12. 20. Every plane rigure bounded by four straight lines is called a QUADRILATERAL, and the right line joining the opposite angles is called a diagonal. 21. A PARALLELOGRAM is a quadrilateral of which the opposite fides are parallel. Fig. 13. 22. A RECTANGLE is a parallelogram which has all its angles right angles. Fig. 14. 23. A SQUARE is a parallelogram which has all its fides equal and all its angles right. Fig. 15. 24. A RHOMBUS is a parallelogram which has all its fides equal. Fig. 17 25. A TRAPEZIUM is a quadrilateral which has not its oppofite fides parallel. Fig. 18. 26. A TRAPEZOID is a quadrilateral which has two of its oppofite fides parallel. Fig. 19. 27. Plane figures bounded by more than four straight lines are called POLYGONS. Fig. 16. 28. A PENTAGON is a polygon of five fides, a HEXAGON hath fix fides, a HEPTAGON seven; an OCTAGON eight; a NONAGON nine; a DECAGON ten; an UNDECAGON eleven; and a DODECAGON hath twelve fides. 29. A REGULAR POLYGON hath all its fides, and all his angles equal; if they are not equal, the polygon is IRREGULAR. 30. A CIRCLE is a plane figure bounded by one line called the circumference, which is such that all ftraight lines drawn to it from a certain point within it called the centre are equal; and these straight lines are called the radii of the circle. The circumference itself is alfo often called a circle. Fig. 20. 1. The DIAMETER of a circle is a straight line paffing through the centre, and terminated both ways by the circumference. 32. An Arc of a circle is any part of its circumference. Fig. 21. 33. A CHORD is a straight line joning the extremities of an arc. Fig. 21. 34. A SEGMENT is any part of a circle bounded by an arc and its chord. Fig. 21. 35. A SEMICIRCLE is half the circle, or a fegment cut off by a diameter. The half circumfe rence is also sometimes called a femicircle. Fig. 20. 36. A SECTOR is any part of a circle which is bounded by an arc, and two radii drawn to its circumference. Fig. 22. 37. A QUADRANT, or quarter of a circle, is a sector having a quarter of a circle for its arc and its two radii are perpendicular to each other. A quarter of the circumference is also called a quadrant. Fig. 23. 38. The HEIGHT OF ALTITUDE of a figure is a perpendicular let fall from an angle or its vertex to the oppofite fide or base. Fig. 24. 39. In a right angled triangle the fide oppofite the right angle is called the HYPOTHENUSE, and the other two fides are called the LEGS, or fometimes the base and perpendicular. Fig. 10. 40. The circumference of every circle is suppofed to be divided into 360 equal parts called DEGREES, and each degree into 60 MINUTES, each minute into 60 SECONDS, and so on. Hence a femicircle contains 180 degres, and a quadrant 90 degrees. 41. The MEASURE of a RECTILINEAL ANGLE is an arc of any circle contained between the two lines which form that angle, the angular point being the centre, and it is estimated by the number of degrees in that arc. Fig. 25. 42. IDENTICAL FIGURES are such as have all the fides and all the angles of the one, respectively equal to all the fides and all the angles of the other, each to each, so that if the one figure were applied to, or laid upon the other, all the fides of the one would exactly fall upon and cover all the fides of the other, the two becoming as it were but one and the fame figure. 43 The DISTANCE of a POINT from a LINE is the straight line drawn from that point perpendicular to, and terminating in that line. 44. An ANGLE in a SEGMENT of a CIRCLE IS that which is contained by two lines drawn from any point in the arc of the segment to the extremities of that arc. Fig. 26. 45. An ANGLE on a SEGMENT, or an ARC, is that which is contained by two lines drawn from any point in the opposite, or fupplemental part of the circumference, to the extremities of the arc, and containing the arc between them. Fig. 26. 46. An ANGLE at the CIRCUMFERENCE is that whose angular point is any where in the circumference, and an angle at the centre is that whose angular point is at the centre. Fig. 26. 47. A TANGENT to a CIRCLE is a straight line that meets the circle at one point, and every where else falls without it. Fig. 27. 48. A SECANT is a straight line that cuts the circle lying partly within and partly without it. Fig. 27. 49. A RIGHT LINED FIGURE is inscribed in a circle, or the circle circumscribes it when all the an angular points of the figure are in the circumference of the circle. Fig. 28. 50. A RIGHT LINED FIGURE circumscribes a circle, or the circle is infcribed in it when all the fides of the figure touch the circumference of the circle. Fig. 28. 51. ONE RIGHT LINED FIGURE is inscribed in another, or the latter circumscribes the former when all the angular points of the former are piaced in the fides of the latter. Fig. 28. 52. SIMILAR FIGURES are those that have all the angles of the one equal to all the angles of the other, each to each, and the fides about these angles proportional. 53. The PRIEMETER of a FIGURE is the sum of all its fides taken together. Note. When the word line occurs, without the addition of either Araight or curved, a straight line is always meant; also the contractions (Def.) (Ακ.) (Th.) are references to the definitions, axioms and theorems that have been before mentioned. AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. When equals are added to equals, the wholes are equal. 3. When equals are taken from equals, the remainders are equal. 4. When equals are added to unequals, the wholes are unequal. 5. When equals are taken from unequals, the Aremainders are unequal. 6. Things which are doubles of the same thing are equal to one another. Things which are halves of the fame thing are equal. 8. The whole is equal to all its parts taken to gether. 6. Things which coincide, or fill the same space are identical, or mutually equal in their parts. 11. Angles that have equal measures, or arcs, a're equal. 12. More than one straight line cannot be drawn from any given point to another given point. Fig. 1, 13. If the perpendicular distance of two points D, and F in a line MN, from another line AB in the fame plane be unequal, the lines AB and MN when indefinitely produced will meet on the fide of the least distance. Fig. 29. REMARKS. A PROPOSITION is something proposed to be done, and is either a Problem or Theorem. A PROBLEM is a thing proposed to be done. A THEOREM is something proposed to be demonftrated. A LEMMA is something premised or demonstrated, in order to make what follows the more easy. A COROLLARY is a consequent truth gained immediately from some preceding truth or demonstration. A SCHOLIUM is a remark or observation made upon something going before. THEOREM І. fig. 30. If two triangles have two fides and the encluded angle of the one equal to two fides and the included angle of the other, the triangles will be identical, or equal in all respects. In the two triangles ABC, DEF, if the fide AB of the one be equal to the fide DE of the other, and the side AG equal to DF. alfo the angle A equal to the angle D, the triangles shall be identical, or equal in all respects. For conceive the triangle ABC, to be applied to, or placed upon the triangle DEF, so that the point A may coir. cide with D, and the fide AB with the fide DE, then since the angles A and D are equal, the fide AC shall alfo coincide with DF, and fince AB is equal to DE, and AC is equal to DF, the points B and E shall coincide, as also the points C and F; confequently the fide BC will coincide with the fide EF; (Ax. 12.) therefore the two triangles are identical (Ax. 9.) and have all their other corresponding parts equal. THEOR. II. fig. 30. Triangles which have two anglesand the fide which lies between them equal are identical, or have their other fides and angles equal. Let the two triangles ABC, DEF have the angle B equal to the angle E, the angle C equal to the angle F, and the fide BC equal to the fide EF, then these triangles will be identical. For conceive the triangle ABC placed upon DEF, fo that BC may fall exactly exa upon EF, then fince the angle B is equal to the angle E, the fide BA will fall upon DE, and in like manner because the angles C and F, are equal, the fide CA will fall upon FD, thus the triangles will exactly coincide and therefore (Ax. 9.) are identical. THEOR. III. fig. 31. In an isoiceles triangle the angles at the base are equal. If the triangle ABC be isosceles, or have a fide AB equal to a fide AC; then will the angle at B be equal to the angle at C. For conceive the angle at A to be bisected, or divided into two equal parts by the line AD. Then the triangles BAD, CAD having two fides and a contained angle of the one equal to two fides and the contained angle of the other, namely A B equal to AC, and AD common to both, and the angle BAD equal to the angle CAD, are equal in all respects (Th. r.) therefore the angle B is equal to the angle C. COROLLARY I. An equilateral triangle is also equiangular. COR. 2. A line that bisects the vertical angle of an isosceles triangle, bisects also the bafe, and is perpendicular to it. THEOR. IV. fig. 31. If a triangle have two of its angles equal, the fides which fubtend or lie opposite to these angles are also equal. Let ABC be a triangle, of which the angles at B and C are equal, the fide AB will be equal to the fide AC. Suppose BC to be bisected at D, and AD joined, dividing the triangle ABC into the two triangles BAD, CAD; and conceive the triangle ABD to be turned over, so that the line DB may fall upon DC, then the point B will fall upon C, and fince the angles B and Care equal, the fides BA fall upon CA, and the extremity C of the fide AC will coincide with the extremity Cof the fide BC, because DC is common to both ; confequently the fideAC is equal to the de BC. COR COR. Hence every equiangular triangle is also equilateral. THEOR. V. fig. 32. Triangles which have their three fides mutually equal, are identical, or have all their three angles equal each to each. Let the triangles ABC, ADB have their 3 fides mutually equal, viz. AB equal to AB, AC equal to AD, and BC equal to DB, the angles oppofite to these fides shall be equal, namely BAC to DAB, BCA to BDA, and ABC to ABD. Suppose the triangles joined by their longest equal fides, and join CD. Then the angle ACD is equal to ADC and the angle CDB to the angle BCD (Th. 3.) therefore the whole angle ACB is equal to the whole angle ADB (Ax 2.) and fiuce AC is equal to AD, alfo BC to DB, the triangles ABC, ADC are identical. (Th. 1.) THEOR. VI. fig. 33. The angles which one line makes with another upon one fide of it are together equal to two right angles. Let the line AB make with CD upon one fide of it, the angles ABC, ABD, these are together equal to two right angles. If AB be perpendicu lar to CD the angles ABC, ABD are evidently equal to two right angles (def. 7.) But if AB be not perpendicular to CD, draw BE perpendicular to CD, dividing the greater angle ABC into the two angles EBC, EBA, then the former EBC being a right angle, and the remaining part EBA together with the whole leffer angle ABD equal to another right angle, the whole of both the proposed angles muft neceffarily be equal to two right angles. (Ax. 2.) COR. I. Hence also, conversely, if the two angles ABC, ABD on both sides of line AB make up together two right angles then CB and BD form one continued straight line. COR. 2. All the angles that can be made round a point by any number of lines are equal to four right angles. THEOR. VII. fig. 34. If two lines interfect each other, the opposite angles are equal. Let AB and CD interfect each other in E, the angle AEC is equal to BED, and AED to BEC; For the angles AEC, AED are together equal to two right angles (Th. 6.) and in like manner BED, AED are equal to two right angles; therefore the angles AEC, AED are together equal to BED, AED (Ax. 1.) and taking away the common angle AED from both, there remains AEC equal to BED (Ax. 3.) In like manner it will appear that AED is equal to BEC. THEOR. VIII. fig. 35. Two straight lines perpendicular to one and the same straight line are parallel to each other. Let AB, CD be perpendicular to EF, the lines AB, CD are parallel. For if they be not parallel let them meet at some point, as G, take EH equal to EG, and join FH. The triangles EHF, EGF having EH equal to EG and EF common to both triangles, and also the angles FEH, FEG e qual, are equal in all refpects, (Th. 1.) and so the angles EFH, EFD being both right angles, the line HFDG, as well as HEG, must be one continued straight line; (Th. 6. Cor. 1.), which is impoffible, therefore AB and CD are parallel. THEOR. IX. fig. 36. If two straight lines be parallel, the perpendiculars to the one terminated by the other, are equal, and are also perpendicular to both the parallels. Let AB and CD be parallel straight lines, and let EF, GH, perpendiculars to AB one of them at E and G, meet the other at Fand H; the lines EF and GH are equal between themselves, and also perpendicular to CD. It is evident that EF and GII are equal, for if they were not equal, AB would not be parallel to CD. (Ax. 13.) The line EF must also be perpendicular to CD, for if it be not, then draw FM perpendicular to FE, meeting GH in M; fo shall FM be parallel to AB (Th. 8.) and therefore GM equal to EF, or to GH, which is impoffible; therefore EF is perpendicular to CD, and by the fame argument GH is perpendicular to CD. : THEOR. X. fig. 37. If a line interfect two parallel lines, it makes the alternate angles equal. Let the line EF interfect the parallel lines AB, CD at Gand H, the alternate angles AGH, GHD are equal. Let HK, GL be perpendicular to the parallel lines AB, CD, then these lines HK, GL are alfo parallel, (Th. 8.) now the triangles HKG, HGL having the fide HK equal to GL and KG equal to HL (Th. 9.) also the angles at K and L equal, they being right angles, will have the angles KGH, LHG equal. (Th. r.) Cor. If a line interfect two parallel lines it makes the exterior angle equal to the interior and oppofite on the fame fide, and also the two interior angles on the fame fide equal to two right angles. For the interior angle GHD is equal to AGH, that is, (Th. 7.) to the exterior angle EGB, to each of these add BGH, and the two interior angles BGH, GHD are together equal to BGH, BGE, that is to two right angles. (Th. 6.1 THEOR. XI. fig. 38. If a line intersecting two other lines make the alternate angles equal, these lines are parallel. Let EF interfect the lines AB, CD at G and H, and make the alternate angles AGH, GHD equal, the lines AB, CD are parallel. For if AB or AG be not parallel to CD, fuppofe KG parallel to CD, then the angle KGH will be equal to GHD, (Th. ro.) that is by hypothefis to AGH which is impoffible, (Ax. 8.) therefore no other line than AB can be parallel to CD. COR. If a line intersecting two other lines makes the exterior angle equal to the interior angle on the same side, or the two interior angles on the fame fide equal to two right angles, these lines are parallel. THEOR. XII. fig. 39. If one fide of a triangle he produced, the exterior angle is equal to both the interior and opposite angles, and the three in-. terior angles are equal to two right angles. Let BC a fide of the triangle ABC be produced to D, the exterior angle ACD is equal to the two interior and opposite angles BAC, ABC, and the three interior angles ABC, BAC, BCA are equal to two right angles. Let CE be parallel to AB, then the angle ACE is equal CAB (Th. 10.) and the angle ABC to ECD, (Th. 10, Cor.) therefore the angle ACD is equal to the two angles CAB CBA, to each of thefe equals add ACB, thus the angles ACB, ACD are equal to the three angles ABC, CBA, BAC, but ACB, ACD are equal to two two right angles (Th. 6.) therefore the three angles of the triangle are equal to two right angles. COR. I. The exterior angle of a triangle is greater than either of the interior opposite angles. COR. 2. Any two angles of a triangle are together less than two right angles. COR. 3. If two triangles have two angles of the one equal to two angles of the other, the remaining angle of the one is equal to the remaining angle of the other. COR. 4. The two acute angles of a right angled triangle are together equal to a right angle. THEOR. XIII. fig. 40. The greatest fide of every triangle fubtends the greatest angle. Let ABC be a triangle of which the side AB is greater than AC, the angle ACB is greater than ABC. Take AD equal to AC and join DC, then the angle ACD is equal to ADC (Th. 3.), but ADC is greater than ABC (Th. 12. Cor. 1.) therefore ACD is greater than ABC, much more then is ACB greater than ABC. Cor. The greatest angle of every triangle is fubtended by the greatest fide. THEOR. XIV. fig. 41. The opposite sides and opposite angles of a parallelogram are equal, and the diagonal divides the parallelogram into two e qual parts. Let BC be a parallelogram, AB is equal to CD, and AC to BD, also the angle CAB is equal to CDB, and ACD to ADB, and the triangle ACD is equal to ABD. For fince AB is parallel to CD (def 21.) the angles BAD, CDA are equal (Th. ro.) and since AC is parallel to BD, for the fame reason, the angles CAD, BDA are equal, now AD is common to the triangles ABD, ACD therefore these triangles are identical, (Th. 2.) hence AB is equal to CD, AC to BD, the angle ACD to ABD, the angle CAD to ADB, and BAD to ADC, and consequently the whole angle CAB to the whole angle CDB. THEOR. XV. fig. 41. The lines which join the extremities of equal and parallel lines towards the fame parts are themselves equal and parallel. Let AB be equal and parallel to CD, then AC and BD which hjoin their extremities towards the fame parts are also equal and parallel. Join AD, then the angles BAD, CDA are equal, (Th. 10.) and fince AB is equal to CD and AD common to the triangles ABD, ACD, these triangles are equal in all respects (Th. 1.), therefore AC is equal to BD, and the angle CAD to ADB, hence AC is also parallel to BD. (Th. 11.) THEOR. XVI. fig. 42. Parallelograms standing upon the same base and between the fame parallels are equal. mains the parallelogram EBCF, therefore these parallelograms are equal to one another. COR! 1. Hence triangles standing upon the same base and between the same parallels are ed qual to one another. For let BAC, BEC be two triangles standing on the fame base BC and between the parallels AF, BC, it is evident that they are the halves of the parallelograms BADC, BEFC, and therefore equal. COR. 2. Hence if a triangle and parallelogram stand on the fame base, the triangle is half of the parallelogram. COR. 3. Therefore all párallelograms or trian gles whatever whose bafes and altitudes are equal, are also equal among themselves. THEOR. XVII. fig. 43. The complements of a parallelogram are equal. Let BD the diagonal of a parallelogram ABCD be drawn, and let HK, EG parallels to its fides interfect each other at F a point in the diagonal; the whole parallelogram is thus divided into four parallelograms; two of these, viz. EK, HG stand about the diameter, and the remaining two HE, GK are called the complements, and are to be proved equal. The whole triangle DAB is equal to the whole triangle DCB, (Th. 14.) and for the same reason the parts DEF, FHB are respectively equal to the parts DKF, FGB, therefore the remaining parts HE, GK, must likewife be equal. THEOR. XVIII. fig. 44. In a right angled tri angle the square of the hypothenuse is equal to the sum of the squares upon the other two fides. Let AD be a square upon the hypothenuse of a right angled triangle ABC and BG, BI squares upon its fides, AD is equal to the fum of BG and BI. Let MBH be parallel to AE meeting GF, produced in H, and let EA, produced, meet GH in N. If from the equal angles GAB, CAN the angle NAB, common to both, be taken away, there remains NAG, equal to BAC, now the angle AGN is equal to ABC, and the fide AG is equal to AB, therefore AN is equal to AC (Th. 2.) or to AE and therefore the parallelograms AM, AH, are equal (Th. 16. Cor. 3.) but AH is equal to the square BG (Th. 16.) therefore AM is equal to BG, and in the same way it will appear that CM is equal to the square BI, therefore the whole square AD is equal to the fum of the squares BG, and BI. THEOR. XIX. fig. 45. A perpendicular drawn from the centre of a circle to a chord bisects that chord. Let CD be drawn from the centre C perpendi cular to AB a chord in the circle, AD is equal to DB. Join CA, CB. Because AC is equal to CB (def. 3.) the angles CAB, CBA are equal (Th. 3.) now ADC, BDC are equal, being right angles, therefore the angles ACD, BCD are equal, (Th. 12. Cor. 3.) therefore the triangles ACD, BCD are in all respects equal, (Th. 1.) and confequently AD-equal to DB. Let ABCD, EBCF be parallelograms standing on the same base BC, and between the fame parallels BC, AF, they are equal to one another. For fince AD is equal to BC, that is to EF, (Th. 14.) therefore AE is equal to DF, now AB is equal to DC (Th. 44.) and the angle BAE to CDF, (Th. ro. Cor.) therefore the triangles BAE and CDF are equal. (Th. 1.) Now if from the whole figure BAFC there be taken away the triangle CDF, there remains the parallelogram ABCD, and if from the same figure there be THEOR. XX. fig. 46. A straight line drawn taken away the equal triangle BAE, there re- through any point in the circumference of a circle, COR. A perpendicular bisecting any chord at right angles pattes through the centre of the circle |