figure without changing the length of the side. This will be best illustrated by forming a square and rectangle with slips of wood joined by a nail at each corner, and then pulling it by the opposite corners so as to make it change its form. Polygon, from pollos, many, and gonia, an angle, and literally signifies a figure having many angles. Proposition, from propositio, something proposed, a thesis. or subject to be discussed. Problem, from problema, a question, or something proposed to be done. Theorem, from theorema, something which we contemplate: hence a theorem is more speculative, a problem more practical. Demonstration, from demonstratio, a pointing out clearly. Corollary, from corollarium, a corolla or coronet, or reward given to actors, champions, &c., above their due : thence it is something over and above the proof of the proposition, an accession to it. Having made these observations on the principles of geometry, you may answer the following questions, taken from Dr. Ritchie's Principles of Geometry. Is arithmetic a branch of mathematical science? Yes: arithmetic is one of the two great branches into which the science of mathematics is divided. Give an example of a problem in arithmetic. If the sum and difference of two numbers be given, and it be required to find the numbers, we have what is called an arithmetical problem. To divide 20 into two parts which shall be to each other as 2 to 3 is a problem which is solved thus: 5 is: 2 :: 20: 8 and 5: 3 :: 20:12 Give an example of a theorem in arithmetic. If we assert that the greater of two numbers equals half their sum, minus half their difference, we have what is called a theorem. Any number divided by 9 will leave the same remainder as the sum of its digits divided by 9. Thus 384 divided by 9 leaves a remainder of 6, and 3 added to 8, added to 4, or 15 divided by 9, leaves a remainder of 6. Give examples of a problem and theorem in geometry. If it be required to find two straight lines whose sum and difference are given, we have a geometrical problem, and if we say that the greater of two straight lines equals half their sum minus half their difference to make an angle equal to a given angle. Does the magnitude of an angle depend on the length of its sides? The magnitude of an angle does not depend on the length of its sides, but on the opening between them. Can two straight lines drawn from the same point inclose a space? It is obvious that two straight lines drawn from the same point cannot inclose a space, otherwise one of the lines must first go from the other, and then return and cut it, which is contrary to the idea of a straight line. If you consider a solid a die for playing, by what is it bounded? A solid is bounded by surfaces. What is the boundary of a surface? A surface is bounded by lines. What is it which divides one surface from that which is adjacent to it? A line either straight, or crooked, or curved. How many points are necessary for determining the position of a straight line? Two points are absolutely necessary to determine the position of a straight line. Since you are only allowed the use of a pair of compasses and a straight scale, by how many methods can you determine the position of a point by the intersection of one straight line with another? By the intersection of two straight lines, or of a straight line with the circumference of a circle, or by the intersection of the circumference of two circles. What is the meaning of the terms complement and supplement? The term complement signifies the angular space by which an angle falls short of a right angie, and the term supplement means the angular space by which an angle falls short of two right angles; as all angles are measured by their subtending arcs, we have arc + complement, equal 90°; an arc + supplement, equal 180°. What are the conditions which render triangles equal in every respect? When two sides and the contained angle in the one is equal to two sides and the contained angle in the other. When are triangles said to be of the same affection? When they are both acute-angled, both rightangled, or both obtuse-angled. Does the proof or demonstration of the preceding theorems, called demonstration by superposition, depend on the accuracy with which the triangles are constructed? The demonstration does not depend on the accuracy of construction, for we can conceive the triangles placed above each other without actually constructing them. When you have two equal or identical triangles, by what method do you discover those angles in the one which are equal to certain angles in the other? Those angles are equal which are opposite to equal sides. When does an angle become equal to two right angles? An angle becomes equal to two right angles when it is measured by an arc of 180°; or when the sides or legs of the angle are in the same straight line. For any angular space less than this is less than two right angles; and any angular space greater than this is greater than two right angles. A right angle contains ninety degrees of angular space. An acute angle must be greater than 0 degree and less than a right angle; and an obtuse angle must be greater than one right angle, and less than two. But an angle generally speaking may contain any number of angular degrees or any part thereof whatever. Are the relative sizes of different countries on the surface of the globe accurately represented on the plane surface of a map? They are not represented with perfect accuracy, but sufficiently near for ordinary purposes. When you speak of turning a triangle round one of its sides as on a hinge, is this necessary for the demonstration of the property, or is it merely used to aid the imagination of the learner? It is merely to aid the imagination of the learner. What do you understand by the distance between two trees? It is the line intercepted between the centres of their horizontal sections. What do you understand by the distance of a point from a given straight line? It is the perpendicular drawn from that point to the straight line. What do you understand by the limb of an instrument for measuring angles? It is the outer rim or circumference thereof. What do you understand by an angle at the centre of a circle? An angle having its angular point in the centre and standing on a given arc. What by an angle at the circumference? An angle having its angular point in the circumference and standing on a given arc. What is a tangent to a circle? A line drawn from a point in the circumference, and which does not cut the circle. When is one circle said to touch another? When they meet in a point but do not cut each other. What is meant by the term analysis? It signifies taking to pieces, or separating the parts of a whole. What is meant by the term synthesis? It is the opposite of analysis, and signifies putting together the parts and thereby making one primary of the whole. If you attempt to instruct a pupil younger than yourself, whether would you employ the analytical or synthetical process of reasoning? The analytical method of solving a problem is to be preferred to the synthetical. Whether is an arc or its chord the longest? The arc is longer than its chord. What is meant by a rectilineal figure inscribed in a circle? It is a right lined figure having all its sides contained wholly within the circle, and its extremities meeting the circumference. What is meant by the circumscribing figure? A circumscribing figure is one which surrounds another, but meets it in certain parts. Whether is the sum of the sides of an inscribed square greater or less than that of the sides of the octagon? The sum of the sides of an inscribed square is less than that of the sides of the inscribed octagon. Whether is the circumference of a circle greater or less than six times its radius or three times its diameter? The circumference of a circle is greater than six times the radius or three times the diameter. By continually doubling the sides of a regular polygon inscribed in a circle, does the perimeter or sum of the sides continually approach to the length of the circumference? Yes; the sum of its sides may be made to approach nearer and nearer to the length of the circumference. Can the perimeter of an inscribed polygon ever exceed the length of the circumference? It can never exceed the length of the circumference. What do you mean by the length of the circumference? I mean the length of a straight line which if bent round the circle would coincide with the circumference in every point. Is the circumference of a circle therefore the limit to which the perimeter of the inscribed polygon continually approaches when the number of its sides is increased indefinitely? The circumference of a circle is therefore the limit to which the perimeter of the inscribed polygon continually approaches when the number of its sides is increased indefinitely. |