To find the dimensions of a standard gallon measure. Pupil's rate of advance. Wall crane; surface; circle; plane. Shear-legs; degrees of freedom; conditions that Various approaches to the fact that the three angles of Plans of playground to various scales by chain-surveying; comparison of plans with one another and with play- ground. Plans by means of plane table; comparison; meaning of ratio. Horse-dealing; index notation; property of indices. Comparison of ratios involving incommen- surable numbers. Conditions that fix the shape of a Morse Code; summation of series. Semaphore signalling. Flag signalling. Rosettes; combinations of n things p at Perpendicular and slant distances of a point from a line. The greater side of a triangle is opposite the greater angle; Table of powers of 2 used to avoid multiplication; powers of 1.01 more useful. Table of powers replaced by graph. Graph replaced by graduated line; slide rule. Laws of Danger angle; the angle in a segment of a circle is con- Index-power table with 1.001 as base; advantages of base 1.0023. Negative and fractional indices. Logarithms with CHAPTER XV. THEOREM OF PYTHAGORAS. Theorem arrived at in various ways; converse. Determina- tion of a right-angled triangle from various data. Quadratic equations. Calculation of right-angled triangle. Condi- tions that fix shape of right-angled triangle. To find right- CHAPTER XVI. CALCULATION OF A TRIANGLE. Calculation from a A B; trigonometrical relations; case of LB obtuse; sine and cosine of an obtuse angle. Calcula- Existence of perpendicular to a plane. Perpendiculars to a plane are parallel; converse. Number of points that two planes. Position of electric lamp fixed by lengths of three supporting strings; discussion of the figure by draw- ing and by calculation. Volume of a pyramid. Position of solid body fixed by six conditions. Conditions that fix a framework in form and in position. Congruence of A SCHOOL COURSE OF MATHEMATICS CHAPTER I THE CACHE 1. A BOY Sometimes makes in a garden a cache or hidingplace for treasures, and covers it over so as to be indistinguishable. By what measurements can he fix the spot so as to find it again? The pupils should be asked for suggestions, and their proposals discussed till a possible method is found. If the discussion can take place in a garden, it will go better. They will suggest measuring its distance from trees, posts, or other known points. Suppose the cache C is 7 feet from an apple-tree A, 12 feet from a plum-tree P, 4 feet from a cherry-tree T, and so on. Mark all the 2. The cache is 7 feet from the tree A. points at this distance from A; in the garden if possible, with a piece of string, or, failing that, on paper with string or compasses, using 1 centimetre to represent 1 foot. These points make a curve that is called a circle, the apple-tree is at the centre of the circle, and the distance of every point of the circle from the apple-tree (7 feet) is called the radius of the circle. The cache lies somewhere on this circle. Now take the next measurement, 12 feet from the plum-tree P, and mark all the points that satisfy this condition, that is, all the points that are 12 feet from the plum-tree. What kind of curve does this give? What is its centre? and its radius? The cache lies on this line also. From these two measurements what do you know of the position of the cache ?-It must be at one of the points where these two curves cross, and we could find it by digging at these points. But how could you avoid digging at the wrong point? We could use another measurement. Any other way?-We could note that the cache lies north or east of the line joining the apple- and plum-trees, or note in some other way on which side of the line it lies. 3. Suppose the trees A and P to be 9 feet apart, and the cache 11 feet from A and 7 feet from P. Mark the trees on paper, 1 cm. representing 1 foot, and mark the two possible positions of the cache. With the same two trees, and the cache supposed to be 10 feet from A and 8 feet from P, mark its possible positions. Suppose that the boy noted the position by fastening one end of a string at A, stretching it to C and making a knot there and fastening the string there, and stretching it from there to P, and cutting it off at P. How could he use this string to find the cache? Do it, taking the trees 9 feet apart and the two parts of the string 11 and 7 feet long. How many points might he find if he forgot whether 11 feet was the distance from A or from P? Mark them all. Suppose that when he tried to use the string he found the knot had slipped, what would he know of the position of the cache? Use the same lengths as before, and mark all the positions he would get by supposing the knot at different points along the string. Use the string or your compasses, whichever you like. 4. Are there any positions of the knot on the string that will not do?-Clearly 1 foot from an end will not do. How does it fail? Distinguish between the positions that will do and the positions that won't. (The pupils will find by experiment that the knot must be at least 4.5 feet from an end.) The two distances from the trees have been supposed to make up 18 feet, or in the drawing 18 cm., and we have found that the distance from each tree must be at least 4.5 feet, or in the drawing 4.5 cm. Suppose this restriction that the two distances make up 18 feet or cm. removed, and find by experiment what relation there must be between the two distances of the cache from the trees, the trees being 9 cm. apart on your drawing. Take 3 cm. as the distance from A, and take in succession 3, 4, 5, 6, . . . cm. as the distance from P. Then take 4 cm. as the distance from A, and the same succession of distances from |