35. The DIAMETER of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. 36. An ARC of a circle is any part of the circumference. 37. Magnitudes which coincide the whole of one with the whole of the other, or exactly fill the same space, are said to be equal in every respect. 38. When the several parts of one magnitude can be made to coincide with those of another, the magnitudes are equal only in area, but in other respects they may be very unequal. A PROPOSITION is a distinct portion of abstract science, the verbal statement of which is called its enunciation, and is either a theorem or a problem. A THEOREM is a proposition which requires to be established by a process of reasoning called a demonstration. A PROBLEM proposes some operation to be performed, and requires a solution. A LEMMA is a subsidiary theorem the truth of which must be established preparatory to the demonstration of a subsequent theorem or problem. An AXIOM is a self-evident theorem, which cannot be made more certain by any proof. A POSTULATE is a request made by the geometer that he shall be allowed to perform a simple operation, evidently possible and absolutely necessary for the construction of the diagrams, or it is a self-evident problem. A COROLLARY is an obvious consequence that results from a demonstration. A direct demonstration proceeds by a regular series of deductions from the premises to the conclusion. An indirect demonstration establishes a proposition, by proving that every other possible supposition contrary to the truth enunciated leads to a conclusion that is absurd or contradictory, and has therefore been frequently called a reductio ad absurdum. An HYPOTHESIS is a supposition made in the enunciation of a proposition, or in the course of a demonstration, and may be either true or false. A SCHOLIUM is a remark subjoined to a demonstration on the nature, the application, or the peculiarities of one or more preceding propositions. To join two points is a concise expression for drawing a straight line from the one of them to the other. POSTULATES. 1. Let it be granted that a straight line may be drawn from any one point to any other point. 2. That a straight line may be produced in a straight line to any required length. 3. That a circle may be described from any centre, with a radiant equal to any given straight line. AXIOMS. 1. Things which are equal to the same thing, or to equal things, are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double, triple, etc., of the same thing, or of equal things, are equal to one another. 7. Things which are each one half, one third, etc., of the same thing, or of equal things, are equal to one another. 8. The whole is greater than its part. 9. The whole is equal to all its parts taken together. When in a construction, or in the course of a demonstration, a reference is made to a definition, a postulate, or an axiom, the contractions Def., Post., Ax., are used; and when a proposition already demonstrated is referred to, the number of the proposition is mentioned, as Prop. IV. PROPOSITION I. PROBLEM. FROM a given point to draw a straight line equal to a given straight line. B C Let A be the given point, and BC the given straight line, it is required to draw from the point A a straight line equal to BC. From A as a centre, with a radiant equal to BC, describe (Post. 3) the circle DEF, and from A draw a straight line to any point in the circumference, as E, the line AE is equal to BC; for the straight line AE being drawn from the centre A to the circumference is a radius (Def. 34) of the circle DEF, and therefore equal (Def. 34, Cor.) to the radiant; but the radiant was taken equal to BC, wherefore AE and BC are each equal to the radiant; and things which are equal to the same thing are equal (Ax. 1) to one another, therefore AE is equal to BC. PROPOSITION II. PROB. From the greater of two given straight lines to cut off a part equal to the less. Let AB and C be the two given straight lines, of which AB is the greater, it is required to cut off a part from AB equal to C the less. From A as a centre, with a radiant equal to C, describe (Post. 3) the circle DEF. Because the straight line AE extends from the centre to the circumference it is a radius (Def. 34, Cor.) of the circle DEF, and there A -B fore equal to the radiant; but the radiant is equal to C, wherefore AE is (Ax. 1.) equal to C. From the same point on the same side of a straight line only one perpendicular to it can be drawn. Let the straight line CD be perpendicular to AB at the point C, then any other straight line CE, drawn from C on the same side of AB, cannot be perpendicular to it. A D E B For as by hypothesis CD is perpendicular to AB, the angle ACD is equal (Def. 9) to the angle BCD; but BCD is greater (Ax. 8) than the angle BCE, therefore the angle ACD is greater than BCE. Again, the angle ACE is greater than C (Ax. 8) ACD; but ACD, as already proved, is greater than BCE, much more then is the angle ACE greater than BCE; wherefore the straight line CE does not make equal adjacent angles with AB, and therefore cannot (Def. 9) be perpendicular to it. PROPOSITION IV. THEOR. All right angles are equal to one another. Let CD be at right angles to AB, and GH at right angles to EF, the four right angles which are thus formed at the points C and G are all equal to one another. Fig. 1. Fig. 2. LH K For the angle ACD is equal to BCD, and the angle EGH to FGH; for if they were not, they could not (Def. 9) be right angles. But the angles at C are also equal to the angles at G. For suppose Fig. 1 to be applied to Fig. 2, so that the point C may be on G, and the straight line CA and GE, the line CB shall coincide (Def. A B E G F 3, Cor. 1) with GF, and the line CD shall fall on GH. For if CD do not fall on GH, it must fall either on the one side of GH or on the other, in the position GK or GL. But it cannot fall in the position GK; for then there would be two perpendiculars, GH and GK, to the line EF, at the point G, on the same side of it, which (Prop. 3) is impossible. For the same reason, CD cannot fall on GL. Since therefore CD can neither fall on the one side of GII nor the other, CD must fall upon it; and therefore the right angles at the points C and G are all equal, viz., the angle ACD to EGH, and BCD to FGH. PROPOSITION V. THEOR. The angles which one straight line makes with another on one side of it are either two right angles, or are together equal to two right angles. Let the straight line CD make with AB on the same side of it the angles ACD and BCD, these are either two right angles, or are together equal to right angles. A E B For if the angle ACD be equal to BCD, each of them is (Def. 9) a right angle, and CD is perpendicular to AB. From the point C draw any other straight line CE on the same side of AB. Because CD is perpendicular to AB at the point C, the line CE cannot be (Prop. 3) also perpendicular at the same point to AB on the same side of it; and therefore (Def. 9) the angles ACE and BCE are unequal. Now the angle BCD is equal (Ax. 9) to the two angles BCE and ECD; to each of these equals add the angle ACD, and the two angles BCD and ACD are equal (Ax. 2) to the three BCE, ECD, and DCA; but the angles ECD and DCA are equal to ECA, and, by substitution, the angles ACD and BCD are equal to the angles ACE and BCE; but the angles ACD C |