1. What is QUESTIONS FOR EXAMINATION. geometry ? 2. From whence was the term derived? 3. What was the occasion of the invention of geometry? 4. How is geometry distinguished? 5. What is meant by theoretical geometry? 6. What is the practical geometry? 7. What is meant by elementary geometry? 8. On what does geometry depend? 9. How are the definitions in geometry characterized? How is a point defined? What is the definition of a line? What is a surface? What is a solid ! What is a triangle? How is a circle defined? 10. What is an axiom? Give the examples of axioms. LESSON THE SECOND. 1. Postulates are things required to be granted as true, or possible, before we proceed to demonstrate a proposition. Examples." Let it be granted that a straight drawn from any one point to any other point. straight line may be produced to any length." circle may be described from any centre." line may be "Or that a "Or that a 2. A proposition is when something is proposed either to be done, or to be demonstrated, and is either a problem or a theorem. 3. A problem is something proposed to be done. Example 1. To divide a given line A B fig. 3. into two equal parts. From the points A and B as centres, and with any opening of the compasses greater B than half A B, describe arcs cutting each other in c and d, and draw the line c d, and the point E where the line cd; cuts AB is the middle point required. Ex. 2.-To raise a perpendicular to a given line, C D, at A. C b A d Fig. 4. Take any two equal distances A b, Ad, and from the points b and d with any opening of the compasses greater than bA, describe the arcs cutting each other D in c, and draw the line Ac, which is perpendicular to C D. Example 3.-To bisect the angle B, or to divide it into two equal angles. Fig. 5. B Fig. 5. b From the point B, with any radius, describe the arc A C, and from the points A and C, with the same radius, describe the arcs cutting one another C at b, and draw b B, which will bisect the angle A B C. Example 4.-To describe an equilateral triangle, A B C, that is a triangle whose three sides are equal to a given line. Fig. 6. Fig. 6. A C B Let A B be the given line: from the points A and B, with an opening of the compasses equal to A B, describe the arcs cutting each other in C, and from the point of intersection draw A C, and C B, and the thing is done. Example 5.-To describe a triangle whose sides shall be equal to three given lines, fig. 7. Let the lines be A B C. Fig. 7. AABC с Take B as the base A B, then from A, with an opening of the compassses equal to C, and from B with an opening of the comB passes equal to A, describe the arcs cutting one another in C; draw the lines CA and C B, and the thing is done. Example 6.-Through a given point C to draw a line parallel to a given line, A D. Fig. 8. the points q and C draw a line which will be parallel to A D. Ex. 7.-To describe a square on a given line, A B. Fig. 9. C Fig. 9. A D Raise a perpendicular at each end of the line A B equal to its length, and draw CD, and the thing is done. Example 8.-To find the centre C of any circle. Fig. 10. D E Draw a chord, A B, at pleasure, bisect it in d, with the diameter DE, which diameter being bisected give C as the centre. B Fig. 10. Example 9.-To inscribe a square in a given circle. Fig. 11. D E Draw the diameter A B, and through C draw D E perpendicular to A B, and join A D, B DB, B E, and E A, which will form a square. Fig 11. Example 10.-In a given circle to describe a regular polygon. Divide the circumference of the circle into as many equal parts as there are sides in the polygon, and join the points of division. 4. A theorem is something proposed to be demonstrated. 5. A corollary is a consequent truth, deduced from some preceding truth or demonstration. [We shall give an example or two of theorems.] It is found by mathematical demonstration, (1.) That one line standing upon another makes with it two angles, equal to two right angles. (2.) That if one side of a triangle be produced, the external angle will be equal to both the internal and opposite angles. (3.) That the three angles in every plane triangle are equal to two right angles. These, with the method of bisecting an angle [See p. 171] may be considered as introductory to the famous theorem, commonly known as the "pons assinorum," or asses' bridge, so denominated from its difficulty in the common Elements of Euclid: this is, (4.) The angles at the base of an isosceles triangle, A B C, (that is, of a triangle whose two legs A B and B C are equal) are equal to each other. Fig. 12. E. B Demonstration.-Bisect the angle ABC by the line B D, then the triangle ABD and BDC having the side A B = BC, BD common and the angle ABD = CBD, will also have the angle A equal to the angle C. For if the triangle BCD were to turn on BD as on a hinge, it would be found that it exactly coincided with F the triangle A B D in all its parts. Fig. 12. The corollaries to this theorem are: (1.) That the line which bisects the vertical angle of an isosceles triangle, bisects the base, and is perpendicular to it. (2.) That every equilateral triangle is likewise equiangular. (3.) If the sides of an isosceles triangle be produced to E and F, the angles under the base are equal, that is, E A D = FCD: because the line DA falling upon B E makes two angles, BAD+EAD= = = two right angles. two right angles for the same reason. Taking away therefore B A Ď= BC D, and the remainders EAD and FCD are equal. A Fig. 13. (5.) In any right angled triangle, A B C, the squares upon the sides A B and B C, fig. 13, taken together, are equal to the square on the hypothenuse A C. This is called the Pythagorean theorem, because Pytha. goras is said to have offered to the gods 100 oxen as a sacrifice, in gratitude for the discovery. The geometrical proof of this theorem is too difficult for a work of this kind, we shall therefore substitute an arithmetical solution: suppose the side AC=5, BC= 4, AB = 3, then 5242 + 32, or 25 = 16 + 9, and so it is shewn in the adjoining figure. Corollary. Hence the square upon either of the sides A B, or B C, including the right angle, is equal to the difference of the squares of the hypothenuse and the other side: or equal to a rectangle, contained under the sum and difference of the hypothenuse and the other side: thus 4' 5' 32 or 16 = 25-9 425 x 3 x 5 3 = 8 × 2=16. 6. Upon geometry depend the sciences of trigonometry, mensuration, &c. QUESTIONS FOR EXAMINATION. 1. What are postulates? Give the examples. 2. What is a proposition? 3. What is a problem? How do you divide a given line into two equal parts? How do you erect a perpendicular on a given line. In what manner do you bisect a given angle? How do you describe an equilateral triangle? How do you describe a triangle whose sides shall be equal to three given lines? How do you draw a line parallel to a given line? How do you describe a square on a given right line? How do you find the centre of a circle? How do you inscribe a square in a circle? How do you inscribe a regular polygon in a circle? 4. What is a theorem ? 5. What is a corollary? What is proved respecting one line standing upon another? What is the external angle, made by producing one side, equal to ? What are the three angles in a triangle equal to ? What does the pons-assinorum prove? How is it demonstrated? What are the corollaries deduced from it? What is the Pythagorean theorem? Give the arithmetical proof. What is the corollary deduced from it? 6. What sciences depend on geometry? |