17. If a diagonal of a parallelogram be either equal to a side or less than a side of the same figure, that diagonal is less than the 18. If a rectangular floor be 25 feet 6 inches (25.5 feet) long, and 16 feet 9 inches (16.75 feet) broad, what is its area in square 19. If the length of a rectangular garden be 15 perches, what must be its breadth, so that it may contain an acre? Answ. 20. If the side of a square be 89 inches, what is its area? Answ. 21. Find the side of a square field, containing three acres. Answ. 21.9089 perches. 22. Given the base of a parallelogram equal to 14 chains 32 23. If the base and perpendicular of a triangle be 66 and 32 yards respectively, what is the area? Answ. 1056 square yards. 24. Given the diagonal of a four-sided field equal to 9 chains 62 links, and the perpendiculars to it from the angles which it subtends, equal to 4 chains 20 links, and 5 chains 77 links, respect- ively; to compute the area. Answ. 4a. 3r. 7.3 p. nearly. 25. Given the legs of a right-angled triangle, equal to 2263 and 26. Given the hypotenuse of a right-angled triangle equal to 79 chains 13 links, and one of the legs equal to 58 chains 65 links; to find the other leg. Answ. 53 chains 12 links. 27. The square described on the diagonal of a square is double of the square itself; and a square is double of the square described 28. Given the legs of a right-angled triangle, equal to 136 and 255 inches, respectively; to compute the length of the perpendi- cular from the right angle to the hypotenuse. Answ. 120 inches. 29. Through two given points in two parallels, to draw two straight lines forming a rhombus with the parallels. * The land-surveyor's chain, invented by Gunter of Gresham College, London, is four perches in length, and is divided into 100 equal links. Such being the length of the chain, the area of a square chain is 16 square perches, and accordingly 10 square chains make an acre. To find, therefore, the area of a parallelogram in acres, multiply the length by the perpendicular breadth, taking the chains and links in each as a single number, and from the product cut off five figures as deci- mals. Thus, in the present example, the dimensions are 14.32 and 10-46 chains, and the area is 14:32 × 10-46, or 149.7872 square chains; and by dividing this by 10, we get 14-97872 acres, the same as would be obtained by multiplying 1432 by 1046, and cutting off five figures. The roods are found by multiplying the deci- mal by 4, the roods in an acre, and cutting off five figures; and the perches by multiplying those figures by 40, the square perches in a rood, and cutting off five + Hence, if the side of a square be given, the diagonal will be found by multiply- ing the square root of 2 by the side: and if the diagonal be given, the side will be PREFACE. As the price of the former editions of this work was found to be an obstacle with many who would have wished to employ it as a text-book, the present edition, though with several improvements and additions, is published in a different and a cheaper form. It is done up either in ONE VOLUME, or in TWO PARTS; so that, at the pleasure of the teacher or student, it may be used either as a whole, or one part may be first studied, and after that the other, if the learner wish to obtain a more profound knowledge of elementary geometry. The First Part contains the First Six Books of Euclid, with the Elements of Plane Trigonometry: and the Second comprehends the Eleventh and Twelfth Books of Euclid, with an Appendix in Four Books. This Appendix, besides containing a number of miscellaneous propositions, several of which are curious and valuable, treats of the Tangencies, of Loci, of Porisms, of Isoperimetrical Figures, and of the Quadrature of the Circle; and by means of some of the notes, the student will be made acquainted, to a certain extent, with the nature and character of the ancient Geometrical Analysis. These subjects are necessarily treated of with brevity. What is here given, however, will perhaps be nearly sufficient for the mathematical student of the present day, whose time and attention, after he has acquired a moderate acquaintance with the ancient geometry, will be more profitably devoted to the study of modern science. In addition to what has been already mentioned, the Second Part contains Notes and Illustrations on the preceding part of the work, which will be useful not only to the teacher, but also to the student who wishes to have an extensive knowledge of elementary geometry. In this part also, there are given many exercises, partly new and partly selected, which it may be useful for the student to perform from time to time, when he finds that he has got principles and practice to enable him to do so. In this edition, a number of simple and easy exercises many which are of a practical nature-are prefixed to the First Part, which ought to be performed by every student who wishes to obtain even a moderate knowledge of geometry, and the elementary parts of plane trigonometry: and to the Second Part a short article is subjoined, giving a view of the leading facts regarding the History of Elementary Geometry and Trigonometry. With regard to the Eight Books of Euclid contained in the work, it is proper to state, that Dr. Simson's valuable edition has been generally followed. In the definitions, however, several considerable changes, and, it is hoped, improvements, have been made; and, in various instances throughout the work, repetitions have been done away, that seemed to be unnecessary, and which must often have been embarrassing to the learner. Besides the more lengthened Notes at the end of the work, various foot-notes have been given, which will assist the student, and increase his knowledge. The demonstrations, also, of several propositions have been shortened and simplified, particularly those of the 5th and 35th propositions of the First Book of Euclid; of the 13th and others in the Second Book; of the 14th, 15th, 21st, 31st and 32d of the Third Book; and of several in the remaining Books. The short article on Trigonometry, at the end of the First Part, contains the best methods at present known for the resolution of triangles; and it gives the method of determining the heights and distances of objects in some of the simplest and most useful cases. Throughout the work, indeed, care has been taken to give the matter, directly or indirectly, when opportunities occurred, a practical character, without interfering, however, with the strictness of reasoning, which is so well calculated to train and strengthen the mind of the student, and thus to promote one great object of scientific education. To the Fifth Book a Supplement is annexed, in which the principles of proportion are established in a brief and simple way, and which, for saving the time of the student, may be used with advantage, instead of the method employed by Euclid, which is prolix, and which is generally found by beginners to be somewhat difficult. Of the improvements and additions, some are new, and some are derived from the writings of others. Much originality cannot be expected in a work on a subject, which, for more than twenty centuries, has occupied the attention, and been cultivated by the talents and labour of a great number of eminent men. Besides this, the great object in view in the present publication has been utility; and, accordingly, the writings of others have been carefully consulted, and whatever appeared to be valuable and appropriate, has been freely but not servilely used. Suggestions have been given by several individuals, which have contributed to the improvement of the work. From Mr. Michael Lawler of Dublin, in particular, several judicious remarks have been received, which have been employed with advantage. Glasgow College, Feb. 14, 1845. EXERCISES ON THE SECOND BOOK OF EUCLID. 1. Given the hypotenuse and one leg of a right-angled triangle equal to 2045 and 1924; to find the remaining leg without squaring the given numbers. Answ. 693. 2. If a side of a parallelogram be equal to one of the diagonals, the squares of the two sides which intersect that side are together equal to the difference of the squares of the diagonals. 3. If the sides of a triangle be 2535, 2730, and 2925, respectively, what are the lengths of the segments into which they are severally divided by the perpendiculars from the opposite angles.* Answ, 1485 and 1050, 1755, and 975, and 1638 and 1287. 4. If the sides of a triangle be 8500, 4080, and 5780, what will be the segments of the several sides made by perpendiculars from the opposite angles. Answ. 5236 and 3264, 6800 and -2720, and 7700 and -1920; the mark - denoting that the segment to which it is prefixed is not in the base, but in its continuation. 5. If the base of a triangle be 70, and the other sides 45 and 35; what is the length of the straight line drawn from the vertical angle to the point of bisection of the base? Answ. 20. 6. If two adjacent sides and one of the diagonals of a parallelogram be 100, 120, and 150, what is the length of the other diagonal? Answ. 162.17, nearly. 7. Given the sides of a triangle equal to 21, 17, and 10 feet, respectively; to compute the area. Answ. 84 square feet. EXERCISES ON THE THIRD BOOK OF EUCLID. 1. In a given line, straight or curved, find a point from which, as centre, if a circle be described, it will pass through two given points. When will this be impossible, and when will there be more than one solution? 2. Through one extremity of a given straight line, to draw a perpendicular to it, without producing it. 3. To describe a circle touching a given arc of a circle, and the lines passing through its extremities, and the centre. 4. How many points determine the position of a straight line, and how many the position and magnitude of a circle? 5. If the arc of a segment of a circle be a fourth of the circumference, what is the magnitude of an angle in the segment? 6. If a chord of a circular arc, 17 inches in length, be divided into two parts of 8 and 9 inches, respectively, by another chord, what is the length of the latter, one of its segments being 4 inches? Answ. 22 inches. 7. If the chord of an arc be 336 feet, and the chord of its half 175 feet, what is the diameter of the circle? Answ. 625 feet. 8. If from a point without a circle, two straight lines be drawn This should be solved both by means of the thirteenth proposition and of the fourth corollary to the fifth. The next should be done by means of the twelfth and thirteenth propositions, and the same corollary, cutting it, and if the parts of one of them, without the circle and within it, be respectively 8 inches and 4 inches, while the external part of the other is 6 inches; what is the length of the part of the latter within the circle? Answ. 10 inches. 9. Suppose that, with a view to find the diameter of a circular pond, a point is taken outside, such that tangents from it to the circumference, form with each other an angle of an equilateral triangle, and that the length of each tangent is found to be 12 perches, what is the diameter? Answ. 13·8564 (=8\/3) perches. 10. In a given straight line, to find a point, from which, as centre, if a circle be described with a given radius, it will touch another straight line given in position. When will there be two solutions? When only one? and when will the problem be impossible? 11. From a given point as centre, to describe two circles, each of them touching a given circle. What different cases of this problein may there be? 12. Through a given point, either within or not within a given circle, to draw a straight line cutting the circle, so that the part of it within the circle may be equal to a given straight line. What are the limits of the magnitude of the given line? EXERCISES ON THE FOURTH BOOK OF EUCLID. 1. If a circle be inscribed in a triangle, and if another triangle be formed by joining the points of contact, each angle of the latter triangle is equal to half the sum of the two angles of the original triangle at the extremities of the side on which its vertex stands. 2. In a given circle, to inscribe a quadrilateral having two of its sides and the diagonal drawn through their point of intersection, respectively equal to given straight lines. When will this be impossible, and when will there be only one solution, and when two? 3. Given two opposite sides of a quadrilateral described in a given circle, and one of the diagonals; to construct the figure. What cases may there be? 4. In an equilateral triangle, to describe three circles, each touching the other two, and touching two sides of the triangle. 5. About a square to describe an equilateral triangle. 6. In a square to inscribe an equilateral triangle, having one of its angular points at one of the angles of the square. 7. If two remote sides of a regular pentagon be produced till they meet, the angle which they form is two fifths of a right angle. 8. In a given square, to describe two equal semicircles touching each other, and having their diameters coinciding with two adjacent sides, and each of them touching also one of the remaining sides of the square. 9. If the sides of a triangle be 13, 14, and 15 yards, respectively, what is the radius of its inscribed circle? And what are the radii of the circles, each touching one of the sides and the other two produced? Answ. 4, 101, 12, and 14 yards, respectively. |