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It will have been noticed, that, in the preceding Theory of Ratio and Proportion, the magnitudes compared are assumed to be, what is called, 'commensurable', that is, to have a 'common measure', or common unit of measurement. Now two or more magnitudes are said to have a 'common measure', when each of them contains the unit of measurement a certain number of times exactly without remainder.
Thus two lines, which are 5 yards and 73 yards in length respectively, are commensurable, because, taking the foot as the measure, the first line contains it 16 times, and the second 23 times, exactly. Similarly, two lines which are 2 yards and 1 yards in length respectively, are commensurable, because the first contains an inch 90 times, and the second 54 times, exactly.
But it does not follow, (and in fact it is not true,) that all lines are of this kind, that is, commensurable. Lines, and also areas, have sometimes to be compared, which have no common measure, and are called incommensurable. To these the preceding Theory does not with perfect mathematical accuracy apply, as it does to commensurable magnitudes; although in all such cases a measure may be found which shall approach as nearly as we please to a common measure, and thus render the preceding Theory applicable by approximation, and to all practical purposes sufficiently true.
Euclid's method of treating ratios and proportion, which applies strictly and equally to all magnitudes, commensurable and incommensurable, has not been adopted, simply because it does not admit of being presented in a form sufficiently intelligible to those for whom this little work is designed. It seemed better to employ a method, which, with admitted imperfections, would allure the learner, than to aim at a perfectness of theory, which might lead him either to pass over the subject entirely, or to read it and not understand it.
(1) Define 'ratio'; between what sort of magnitudes can it exist? Is there any ratio' between ten shillings and two miles? If not, why not?
(2) What is the test by which you determine whether, or not, two proposed magnitudes are of the same kind'? Apply it to the case of a triangle and one of its sides. Also to the case of a triangle and a square,
(3) Is there any ratio between an angle and a triangle? Or between a right angle and a square ?
(4) Define the measure of a ratio; and express the ratio of a parallelogram to the triangle' on the same base and between the same parallels'.
(5) What is the ratio of a lineal inch to a lineal yard?
(6) What is the ratio of the square of AB to the square of the half of AB?
(7) Is it necessary, when two lines or magnitudes have a ratio to each other, that the one should contain the other an exact integral number of times? Explain fully.
(8) When is one line, or area, said to be a multiple of another? If 5 times A = 7 times B, what is the ratio of A to B?
(9) Define 'proportion', and 'proportional'. How many magnitudes are concerned in a proportion? May they be all of one kind? Must they be so?
(10) Can two lines and two triangles be in proportion? Can two angles, a triangle, and a parallelogram, be in proportion?
(11) If there be two triangles of equal altitudes, and the base of one be double the base of the other, what is the proportion between the bases and triangles? And what is the ratio of the two triangles?
(12) Shew that, if any two sides of a triangle be bisected, the line joining the points of bisection is parallel to the third side, and equal to half of it.
(13) Define similar triangles; can triangles be similar and not equal? Can they be equal and not similar? Explain fully.
(14) Are all equilateral triangles similar? Are two isosceles triangles necessarily similar?
(15) If each of the sides of a triangle be bisected, shew that the lines joining the points of bisection will divide the triangle into four equal triangles similar to the whole triangle and to each other.
(16) If through the vertex of each angle of a triangle a straight line be drawn parallel to the opposite side, shew that these lines will form a triangle similar to the given triangle; and find the ratio of this triangle to the given triangle.
(17) If the sides of any quadrilateral figure be bisected, shew that the lines joining the points of bisection will form a parallelogram.
(18) Shew that any triangle cut off from an equilateral triangle by a line parallel to one of its sides is equilateral.
(19) Through a given point draw a straight line, terminated by two given straight lines, so that it shall be bisected in that point.
(20) Through a given point draw a straight line, terminated by two other given straight lines, so that it shall be divided by that point in a given ratio.
(21) Of all triangles with two given sides shew that that is the greatest in which the two sides form a right angle.
(22) If an angle of a triangle be bisected by a straight line which also cuts the opposite side, shew that the two parts into which this side is divided will be in the same ratio as the other two sides are to one another.
(23) Shew that any two right-angled triangles are similar, if two of their acute angles, one in each triangle, are equal.
(24) If two triangles have the sides of the one, or sides produced, respectively at right angles to those of
the other, each to each, shew that the triangles are similar.
(25) If each of the sides of a triangle be bisected, and straight lines be drawn from the points of bisection to the vertex of the opposite angle, shew that these three lines will intersect in one point, and that the point of intersection divides each line into two parts of which one is double the other.
(26) In the last problem shew that the three lines from the point of intersection to the vertices of the three angles divide the given triangle into three equal triangles.
(27) Shew that two isosceles triangles will be similar, if any angle of the one be equal to the corresponding angle of the other.
(28) Find the greatest 'mean proportional' between any two lines of given sum.
(29) If two circles touch each other, either internally or externally, and two straight lines be drawn through the point of contact; so as to form four chords, two in each circle, shew that the four chords are proportionals.
(30) If two circles touch each other externally, and a straight line be drawn touching both and terminating at the points of contact, shew that this line is a mean proportional between the diameters.
(31) Shew that the parts into which the diagonals of a trapezium are divided by their point of intersection are proportionals.
(32) Shew that any rectangle is a mean proportional between the squares of two of its adjacent sides.
(33) Shew geometrically that a side of a square and its diagonal are incommensurable'.
(34) If on the sides of a right-angled triangle, taken as bases, three similar rectangles be described, shew that the rectangle on the side opposite to the right angle is equal to the sum of the other two.
POLYGONS, AND THEIR CONNECTION WITH THE
85. DEFINITION. A POLYGON* is a plane surface bounded by more than four straight lines, which are called its sides.
A plane surface with three sides has already received the name of triangle, and with four sides 'parallelogram', 'square', 'quadrilateral, or trapezium', as the case may be; therefore polygons begin with five sides, and may have any greater number.
An angle of a polygon means an angle formed by two adjacent sides of the polygon. And the number of the angles is obviously equal to the number of the sides.
DEF. A Polygon of 5 sides is called a Pentagon,
and so on.
DEF. A Regular Polygon is a polygon which has all its angles equal and all its sides equal.
Thus a regular Pentagon, Hexagon, and Octagon will respectively present the following appearance as to
[It does not yet appear that a regular polygon, as here defined, is a possible construction. All that is meant is, that, if such be possible, these are the distinctive names of such polygons §.]
DEF. The sum of all the sides of a polygon is called its perimeter.
*Polygon, derived from two Greek words, literally means a figure which has many corners.
Pentagon, that is, a five-cornered figure.
Hexagon, that is, a six-cornered figure.
Octagon, that is, an eight-cornered figure.
A similar observation might have been made, when the Definitions of equilateral triangle, and of a square, were given. We were not then able to say, that such constructions were possible.