will form 1 star-shaped polygon; ... ... And diagonals cutting off successively in a 5 and 8, or 6 and 7 sides will form ... ... 2 2 1 5 polygons. A similar process will form other star-shaped polygons. N.B.-The figure as C at the centre of a star-shaped polygon, formed by the intersection of lines from the angular points, 1, 2, 3, &c., is also a regular convex polygon, with the same number of sides as there are re-entrant angles to the star-shaped polygon. REMARKS ON BOOK IV. 1. The Problems of which the Fourth Book of Euclid is entirely composed require little classification; they may be brought under the general heads;- 1st, Of rt.-lined regular figures inscribed in circles; Pr. 2, 6, 11, 15 and 16; 2nd, Of rt.-lined regular figures, circumscribed about circles; Pr. 3, 7, 12; 3rd, Of circles inscribed in regular rt.-lined figures; Pr. 4, 8, 13; 4th, Of circles circumscribed about rt.-lined figues; Pr. 5, 9, 14; 5th, Of constructing an isosceles triangle, having each angle at the base double of the vertical angle, Pr. 10. 2. The Use and Application of the Propositions of this Book have been given so much in extenso, that it would be superfluous to add to these remarks, except again to challenge attention to the value of theoretical reasoning as the guide in Geometry to most important Practical Results. GRADATIONS IN EUCLID. BOOK V. THE THEORY OF PROPORTION, OR OF THE COMPARATIVE MAGNITUDES OF PLANE FIGURES. THIS Book is entirely independent of the four books which preceded it. In the main they relate to the Properties of Figures on a plane surface, but the fifth book introduces Properties of a more general kind, and though restricted by Euclid to the comparative magnitudes of right lines, really extends to all kinds of magnitude. It is no longer absolute equality, or inequality, which we have to consider, but the RATIO, or mode of estimating the relative lengths of lines and the magnitudes of figures, and the proportion, or setting forth of those relative lengths or magnitudes. As in the Second Book material assistance for illustrating the properties of Rectangles, was derived from Algebra and Arithmetic, so in this book similar help will be obtained from the same sources. Indeed very many of the terms employed will be already familiar from their use in Arithmetic; and it will be seen that the estimating of Ratios, and the setting forth of proportions, rest so entirely on a numerical basis, that to a very high degree the Fifth Book, or the Theory of Proportion, is an application of Numbers to the purposes of Plane Geometry. For this reason it will be of advantage to the Learner to be presented with a brief view of the Principles on which are established the Properties of Proportional Numbers and Quantities. If, however, he has already mastered the subject, he may pass over the next few pages, and at once enter on the Theory of Geometrical Proportion. SOME PROPERTIES OF PROPORTIONAL NUMBERS, INTRODUCTORY TO EUCLID'S THEORY OF GEOMETRICAL PROPORTION. We may compare two numbers together, either by their difference, or by their quotient, i. e. by the number of times which the greater contains the less, or the less measures the greater. When we say 12 — 9 - = 3, we compare 12 and 9 by their difference; but when we say 12 contains 9, one and one quarter times, we compare them by a division of 12 into 9 and a part of 9. The proportion which one number or quantity bears to another is often called its Ratio; the ratio, measured by the difference is named an arithmetical ratio, that measured by the quotient,- —a geometrical ratio. Proportion is applied, either to an identity of difference between three or more numbers, as 12, 9, 6, where the common difference or the Arithmetical ratio is 3; or, to identity of relative magnitudes, as 12 : 9 :: 8: 6,-where 12 contains 9 just as often as 8 contains 6,-the common quotient or ratio being 14. When the differences are identical, the numbers are in Arithmetical Proportion; when the contents of each pair of terms are identical, the numbers are in Geometrical Proportion. The term proportion, taken by itself, is usually restricted to numbers in geometrical proportion; and of these we have now to treat. Identity in the quotients of successive pairs of numbers constitutes Proportion. Take for an example, 15:5:: 36: 12; the quotient obtained on dividing 15 by 5 is the same as that obtained by dividing 36 by 12; and these four numbers, 15, 5, 36, and 12,—or any other four numbers fulfilling the condition of equality of quotients in each successive pair, form a Proportion, or set of Proportionals. The extremes are the first and last terms in the series; the terms placed between the first and last terms, the means; the antecedent is the first term of a ratio; the consequent, the second term. The ratio may be expressed, either by a whole number or by a fraction; thus in the proportionals, 18: 6 :: 24: 8, the constant ratio is 3; in the proportionals 12 : 9 :: 36 : 27 that ratio is or 13. a = =) If we take two sets of quantities in direct proportion, a : b:: c d (which may also be written in a fractional form we can readily exhibit various rules that are employed in modifying a Proportion: they are all dependent on the principle that resulting equations are equally true whenever the thing which is done on one side of an equation is also done on the other side. Take as an example α b d abd cbd RULE 1. Multiply each side of the equation by b × d; we obtain = b d or ad = cb; .. the product of the extremes = the product of the a , or = b d i.e., a : b & c : d are proportionals. RULE 3. Add unity to each side, and the equation is b a +1=2&+1; or b c + d i.e., a + b : b = =c+d:d. RULE 4. Subtract unity from each side, we have the equation RULE 5. Any common factor, as m, may be expunged, except from the two extremes, or from the two means; for, if ma = m a d ,on dividing both sides m b or more series of Proportionals, the products of the corresponding antecedents and consequents also constitute a Proportion, and this is named a Compound Proportion; for it on multiplying the terms, on the left hand of the sign, together, and Three or more series of proportionals are compounded in a similar way. The Essential Property, or Criterion, of numbers in true proportion is, that the product of the extremes is equal to the product of the means; and reciprocally, if the product of any two numbers equals the products of any other two numbers, the series of numbers constitutes a proportion; thus, in 15 5 36: 12 ;- 15 × 12 = 180 = 5 × 36; .'. a true proportion. in 14: 5: 35: 12;-14 × 12 = 168; 5 X 35175; .. a false proportion. From the Essential Property of numbers in true proportion it follows, that when any three terms are given the fourth may be found; for on dividing the product of the means by the given extreme, or that of the extremes by the given mean, the other extreme, or the other mean, will be obtained; thus in 6:8; 24: x; If each of the means is the same number, as in 3: 6 :: 6 : 12, the product of the extremes equals the square of one of the means; the value of one of the |