6. 'That if a straight line meet two others, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are together less than two right angles.' THE AXIOMS. We next have a list of nine axioms, in the original kowal Evvoial, common notions, statements of facts which we cannot help knowing, but of which we do sometimes require to be ,reminded. 1. 'Things which are equal to the same thing are equal to one another.' From this naturally follow axioms 6 and 7. 6. Things which are double of the same thing are equal to one another.' 7. 'Things which are halves of the same thing are equal to one another.' These axioms may of course be extended to any multiple or fraction. Axioms 2-5 form a second connected group. 2. If equals be added to equals the wholes are equal.' 3. 'If equals be taken from equals the remainders are equal.' If A=C and B=D, A+B=C÷D. 4. If equals and unequals be added together, the wholes are unequal.' 5. 'If equals be taken from unequals the remainders are unequal.' If A=C but B and D represent unequal magnitudes, then A+B is not equal to C+D. 8. 'Magnitudes which coincide or exactly fill the same space are equal.' 'Things are said to agree together or coincide if when they are applied the one to the other, or set the one upon the other, the one exceedeth not the other in any thing.' A magnitude is that which can be measured. The magnitudes with which we are at present concerned are lines having length-measure, plane figures or superficies having the measures of length and breadth or area, and angles which, as we have seen, are measured by the size of the arc intercepted. This axiom applies to all these magnitudes as well as to others which are not geometrical. 9. The whole is greater than its part.' LECTURE III. ON THE NATURE AND ARRANGEMENT OF A PROPOSITION.- THE subjects treated of in the first book of Euclid are :— (1) The properties of triangles. (2) The theory of parallel straight lines, with the properties of parallelograms. (3) The laws which determine equivalence or equality of area in rectilineal figures. These subjects Euclid discusses in forty-eight propositions (πpóraσes), which are of two kinds, (1) problems in which he shows how something may be done, e.g. how to describe an equilateral triangle, and (2) theorems in which he proves some geometrical law, e.g. the angles at the base of an isosceles triangle are equal. A proposition consists of the enunciation, the construction and the proof. The enunciation states in general terms the problem to be solved or the truth to be demonstrated, e.g. the angles at the base of an isosceles triangle are equal, and (2) applies this general statement to a particular instance; thus let ABC be an isosceles triangle having the side AB equal to the side AC, then the angle ABC shall be equal to the angle ACB. These divisions are sometimes distinguished as the general and particular enunciation. The construction describes the lines and circles, &c., which are to be drawn in order to solve the problem or demonstrate the theorem. Sometimes, however, no construction is required. The proof shows by reasoning that the problem proposed has been solved, or that the theorem is true. There are two methods of proof, 'direct' and 'indirect." A direct proof shows the truth of a proposition directly by means of other truths already established. An indirect proof shows the truth of a proposition negatively, by proving that the truth of any supposition contrary to that in the enunciation being granted, we arrive at untrue or absurd results. The various lines and angles, &c., made use of in a proposition are named from the letters of the alphabet taken consecutively. A line is usually named by two letters placed at its extremities, thus the line AB ab, an angle by three, placing the letter at the angular point in the middle, thus the angle ABC or CBA. A rectilineal figure usually has a letter placed at each angle. There are a few contractions in general use in geometry. Q.E.F. problems are usually concluded by these letters, an abbreviation for 'quod erat faciendum,' which was to be done. Q.E.D. stands for 'quod erat demonstrandum,' which was to be demonstrated, and forms the conclusion of theorems. Hyp Hypothesis. This is used in referring to the supposition involved in the enunciation, e.g. the angles at the base of an isosceles triangle are equal. In this proposition we consider two sides of the triangle to be equal, in accordance with the supposition of an isosceles triangle in the enunciation, that is by hypothesis. The following symbols greatly shorten the labour of writing out demonstrations : = is equal to Zangle rt. right angle ||gram parallelogram A triangle sq. square .. therefore ..since or because PROP. I. Having now cleared the way, we can enter upon the discussion of the first proposition, which is a problem, that is, it shows how something is to be done. We are required to describe an equilateral triangle on a given finite straight line.' 'Describe' is a technical term denoting to draw or construct a mathematical figure. Def. 24 tells us that an equilateral triangle is a figure having three equal sides. The given line will itself form one side of the triangle, and if we can draw two other lines equal to it in such a position that the three lines shall together form an equilateral triangle, the problem will be solved. Draw a line to represent the given line A. B. As the line AB is to form one side of the triangle, its extremities will be the points from which the other two sides must be drawn. We will endeavour to obtain these by the help of the principles already learnt. Of these the postulates are the most likely to be of assistance, since they alone permit us to do anything. We will examine them in order. The first postulate assumes that we may draw a 'straight line from any one point to any other.' Will this help us here? No; it affords no means of measuring the length of the lines which we have to draw. The second postulate, that ‘a terminated straight line may be produced to any length,' fails to help us, for the same reason. We have remaining the third postulate, which assumes that 'a circle may be described from any centre at any distance from that centre.' Let us refer to the definition of a circle (def. 15). It is to the latter part that I wish to draw your attention, 'and is such that all straight lines drawn from a certain point (the centre) within the figure to the circumference are equal.' Here is a mode of measuring the lines required. AB will form a radius of a circle, and if we take A as the centre, we can draw as many lines as we please A B from A to its circumference, which will be equal to AB, In like manner, by drawing a circle, of which B is the centre, with AB again as radius, we shall obtain a measure for lines C |