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EXERCISES ON THE FIFTH BOOK OF EUCLID.
1. If A B C D, prove that mA+B : B :: mC+D: D. 2. On the same hypothesis, prove that mA+nB : pA+qB :: mC+nD: pC+qD.
3. On the same hypothesis, still, prove that A2+B2 : A2—B2 :: C2+D2: C2-Da.
4. Prove also that A2+AB+B2: A2—AB+B2 :: C2+CD+ D2: C2—CD+D3.
5. Find a third proportional to 4 and 6. Answ. 9.
EXERCISES ON THE SIXTH BOOK OF EUCLID.
1. If the sides of a triangle be 532, 684, and 912, respectively, what are the parts into which each of them is divided by the straight line bisecting the opposite angle? Answ. 228 and 304; 252 and 432; and 399 and 513.
2. Given the legs of a right-angled triangle equal to 65 and 156 inches, respectively; to compute the segments into which the hypotenuse is divided by the perpendicular to it from the right angle, and to compute also that perpendicular. Answ. The segments are 25 and 144, and the perpendicular 60.
3. If the base of a triangle be 123, and the other sides 125 and 80, respectively, what is the length of the line bisecting the vertical angle? Answ. 80.
4. If two homologous sides of two similar fields be 15 and 25 perches respectively, and if the content of the greater be 5 a. 1r. 35 p. what is the content of the less? Answ. 1a. 3r. 35 p.
5. If the homologous sides of two similar fields be in the ratio of 3 to 4, and if the one contain two acres more than the other, what are their respective contents? Answ. 2a. 2r. 114p; and 4a. 2r. 11 p.
6. If the joint content of two similar fields be 7 acres, what are their separate contents, their homologous sides being as 5 to 7? Answ. 2a. 1r. 1814p. and 4a. 2r. 2134p.
7. If the base of a triangle be 18, and the other sides 25 and 16, respectively, what is the length of the external segment of the base, made by a straight line bisecting the exterior angle at the vertex? Answ. 32.
8. From the same data find the length of the bisecting line. Answ. 28.28427 (=20√/2).
9. If the sides of a triangle be 13, 14, and 15 inches, respectively, what is the radius of the circumscribed circle? Answ. 8.
10. Through a given point to draw a straight line, forming with two sides of a given triangle, or with one or both of them produced, a triangle similar to the given one. Show that in general this may be done in six different ways, and point out in what circumstances there will be fewer solutions.
EXERCISES IN TRIGONOMETRY.
1. Given the sides of an isosceles triangle, to compute its angles in the simplest manner.
2. Given the angles and any side of an isosceles triangle, to find the remaining sides in the easiest manner.
3. Given the angles of a triangle, and the straight line drawn from the vertical angle to the point of bisection of the base; to compute the sides.
4. Prove that the sine of twice a circular arc, is less than twice its sine; but that the tangent of twice an arc is greater than twice the tangent of the same are; the simple arc in each instance being less than 45°. What varieties will there be in each case, if the arc exceed 45°?
5. Prove the third proposition of the sixth book of Euclid by means of trigonometry.
6. Given two angles of a trapezium, and the three sides which form them; to show the method of computing the remaining side. 7. Compute the angle which a straight line drawn from one angle of a square, to bisect one of the remote sides, makes with that side. Answ. 63° 26', nearly.
8. If the natural sine of an angle be 0.50528, what is its cosine? Answ. 0.86295.
9. If the natural cosine of an angle be 0.86603, what is its cotangent? Answ. 173206.
10. If the natural secant of an angle be 1'4142136, what are its sine and tangent? Answ. 0.7071068 and 1.
11. If the natural tangent of an angle be 0.51872, what is its sine? Answ. 0.46046.
ELEMENTS OF EUCLID.
1. A point is that which has position, but not magnitude. † 2. A line is length without breadth.
Corollary. The extremities of a line are points; and an intersection of one line with another is also a point. ‡
3. If two lines be such that they cannot coincide in any two points, without coinciding altogether, each of them is called a straight line.
Cor. Hence two straight lines cannot enclose a space. Neither can two straight lines have a common segment; that is, they cannot coincide in part without coinciding altogether. §
Definitions explain the precise sense in which terms are to be understood, distinguishing the ideas expressed by those terms from the ideas expressed by any others.
+ In geometrical figures, or, as they are also called diagrams, we are obliged, instead of mathematical points and lines, to employ physical ones; that is, dots, instead of mathematical points, and lines of some perceptible breadth, instead of mathematical ones; as the finest line that we can make has breadth, and the finest point both length and breadth. Our reasoning, however, is not vitiated on this account, as it is conducted on the supposition that the point has no magnitude, and the line no breadth, and nothing in it depends on the magnitude of the point or the breadth of the line in the diagram.
A corollary is an inference which arises easily and immediately from some other principle, not requiring any lengthened process of reasoning to establish its truth.
The truth of this corollary is manifest, since, from the definition of a line, it follows, that the extremities and the intersections of lines have position, but not magnitude, and therefore (definition 1) are points. The corollaries to the fourth and sixth definitions may be illustrated in a similar manner.
This will be illustrated by applying the edges of two straight rulers to one another, and turning them in different ways. It will also be rendered clearer by a negative illustration, by means of two wires, or other substances, bent similarly; as, though these may coincide, entirely when turned in one direction, they will coincide only in some points when placed otherwise.
§ Thus, if ABC, DBC could both be straight lines, they would have the segment or part, BC common. But this is impossible; for if they coincide in any two points, as B and C, the parts AB and DB must also coin
4. A surface, or superficies, is that which has only length and breadth.
Cor. The extremities of surfaces are lines; and an intersection of one surface with another is also a line.
5. A plane surface, or, as it is generally called, a plane, is that in which any two points being taken, the straight line between them lies wholly in that surface.*
Cor. Hence two plane surfaces cannot enclose a space. Neither can two plane surfaces have a common segment.
6. A body, or solid, is that which has length, breadth, and thickness.
Cor. The extremities of a body are surfaces.
7. A rectilineal angle is the mutual inclination of two straight lines, which meet one another. The point in which the straight lines meet is called the vertex of the angle.
8. When one straight line standing on another makes the adjacent angles equal, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it, and is said to be at right angles to it.
9. An obtuse angle is
that which is greater
than a right angle.
10. An acute angle is that which is less than a right angle. 11. Parallel straight lines are those which are
in the same plane, and which, being produced
ever so far both ways, do not meet.
12. A figure is that which is enclosed by one or more boundaries. 13. Rectilineal or rectilinear figures are those which are contained by straight lines:
14. Triangles, by three straight lines:
15. Quadrilateral figures, by four straight lines:
16. Polygons, by more than four straight lines.
* On the contrary, if two points be taken on the surface of a ball, the straight line between them will lie within the ball, and not on its surface. The surface of a ball, therefore, is not a plane surface.
† Or a rectilineal angle is the degree of opening, or divergence of two straight lines which meet one another. A clear idea of the nature of an angle is obtained by gradually opening a carpenter's rule, or a pair of compasses; as the angle made by the parts of the rule or the legs of the compasses, will become greater as the opening widens. It is evident that the magnitude of the angle does not depend on the length of the lines which form it, but merely on their relative positions.
An angle is best named by a single letter placed at its vertex, unless there be more angles than one at the same point. In this case, the angle is generally expressed by three letters, the middle one of which is placed at the vertex, and the others at some other points of the lines containing it. Thus, in the first figure for the fifth proposition of this book, the angle contained by AB and AC is called the angle A, while that which is contained by AB and BC is called the angle ABC.
An equilateral figure is that which has equal sides, and an equiangular one that which has equal angles. A polygon which is equilateral and equiangular is called a regular polygon. Polygons, especially when they are regular, are often distinguished by particular names, derived from the Greek language, denoting the number of their angles, and consequently of their sides. Thus, a polygon of five sides is called a pentagon; of six, a hexagon; of seven, a heptagon; of eight, an ortagon; of nine, an enneagon; of ten, a decagon; of twelve, a dodecagon; and of fifteen, a pentedecagon.
17. Of three-sided figures, an equilateral triangle is that which has its three sides equal:
18. An isosceles triangle is that which has two sides equal:
19. A scalene triangle is that which has all its sides unequal:
20. A right-angled triangle is that which has a right angle:
21. An obtuse-angled triangle is that which has an obtuse angle:
22. An acute-angled triangle is that which has three acute angles.
23. In a figure of four or more sides, a straight line drawn through two remote angles of it, is called a diagonal. ‡
24. Of four-sided figures, a parallelogram is that which has its opposite sides parallel.
25. Any other four-sided figure is called a trapezium.
26. A parallelogram which has a right angle is called a rectangle.†
27. A rectangle which has two adjacent sides equal, is called a square.
28. A parallelogram which has two adjacent sides equal, but its angles not right angles is called a rhombus.§
* From this and the five following definitions, it appears that triangles are divided into three kinds from the relations of their sides, and into three others from those of their angles. When the three sides are equal, the triangle is equilateral; when two are equal, it is isosceles; and when they are all unequal, it is scalene. Again, a triangle may have one right angle, one obtuse angle, or neither one nor other; and hence the distinction into right-angled, obtuse-angled, and acute-angled. It will appear from the 17th or 32d proposition of this book, that a triangle can have only one right or one obtuse angle; and, from the 32d, that it may have three acute angles. It may be remarked that the term scalene is seldom used; and that obtuseangled and acute-angled triangles are often called oblique-angled triangles in contradistinction to right-angled ones.
t In Simson's, and most other editions of Euclid, the diagonals of parallelograms are called their diameters. It is better, however, to confine the term diameter to the lines which are so called in the circle and other curves. It may be remarked, that in this definition, as in many other instances, the term angle is used to denote what in strictness is the vertex of the angle.
A rectangle which is not a square, is sometimes called an oblong.
The following are the definitions of the square and rhombus which are given in Simson's edition: