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The subjoined figure is a Circle, of which the curved line AFGCBD is the circumference; the point E, the centre, from which all lines EA, EF, EC, &c., drawn to the cir. A cumference, are equal. In fact the circle may be supposed to be generated by the revolution of any one of these lines, as EA, round the fixed point E, until it returns to the point from which it began to revolve. Each of these equal lines is called a radius or ray; and is evidently one-half of a diameter. Any diameter, as AB, not only divides the circle into two semicircles, with reference to the equal semi-circumferences ACB, ADB; but the included areas are also manifestly equal.
It may be well to observe that, for geographical and other purposes, the circumferences are supposed to be divided into 360 equal parts, called degrees; and, again, each degree into 60 minutes, each minute into 60 seconds; and so on. These divisions are symbolically expressed thus : 46° 4' 13". Now, in the revolution of the radius AE round the point E, througla one-fourth of the circumference AG, the angle at E will gradually have encreased, till it is subdivided or measured by an arc—so any portion of the circumference is called—of 90°. Again, in the passage from G to B, another arc of 90° is described, and the angle GEB is equal to the angle AEG. Each is therefore a right angle. Angles are said to be of as many degrees as the arcs upon which they stand; and it is clear that, whatever the radius of the circle, the inclination of an angle of any given number of degrees is always the same; so that an angle is not encreased or diminished by the lengthening or shortening of the lines by which it is formed.
19. Rectilineal figures are those which are contained by straight lines.
20. Trilateral figures, or triangles, by three straight lines.
21. Quadrilateral by four straight lines.
22. Multilateral figures, or polygons, by more than four straight lines.
As all rectilineal figures have the same number of angles as sides, they may be named indifferently from either; or sometimes from one, and sometimes the other. Thus, three-sided figures are called triangles, from the Latin tres and angulus; and many-sided figures, polygons, from the Greek words rolùs, many, and ywvía, an angle.
23. Of three-sided figures, an equilateral triangle is that which has three equal sides.
24. An isosceles triangle is that which has only two sides equal.
The word isosceles is derived from the Greek ; and signifies having equal legs, that is sides, from ioos, equal, and okélos, a leg. The base of such a triangle may be either shorter or longer than either of the other sides.
25. A scalene triangle is that which has three unequal sides.
The word scalene is also of Greek origin, signifying unequal.
26. A right-angled triangle is that which has a right angle.
27. An obtuse-angled triangle is that which has an obtuse angle.
28. An acute-angled triangle is that which has three acute angles.
29. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.
30. An oblong is that which has all its angles right angles, but has not all its sides equal.
Called also a rectangle.
31. A rhombus is that which has all its sides equal, but its angles are not right angles.
32. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.
These last four figures come also under the more general definition of a parallelogram; which is given under Prop. XXXIV.
33. All other four-sided figures besides these, are called Trapeziums.
Trapezium is a Greek word, signifying a table.
34. Parallel straight lines are such as are in the same plane; and which, being produced ever so far both ways, do not meet.
Let it be granted,
1. That a straight line may be drawn from any one point to any other point:
2. That a terminated straight line may be produced to any length in a straight line : and
3. That a circle may be described from any centre, at any distance from that centre.
A postulate (from postulare) is a demand so reasonable that it cannot fail to be granted; or, in other words, an assumption of which the grounds are so apparent, as to be admitted without dispute.-To produce is a technical term, signifying to prolong, or extend, from the Latin producere.
It appears from these postulates that the instruments which Euclid places in the hand of the student, are, a straight scale, and a pair of compasses. The instruments employed in practical geometry involve principles investigated in the Elements, which will be noticed as occasions arise.
1. Things which are equal to the same thing, are equal to one another.
Also, things which are equal to equal things, are equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be taken from equals, the remainders are equal.
4. If equals be added to unequals, the wholes are unequal.
5. If equals be taken from unequals, the remainders are unequal.
6. Things which are double of the same, are equal to one another.
7. Things which are halves of the same, are equal to one another.
8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
9. The whole is greater than its part.
10. Two straight lines cannot inclose a space.
In order to enclose a space, two straight lines must have more than one common intersection, which has been already seen to be impossible (Note on Definition 6).
11. All right angles are equal to one another. (See Definition 10.)
12. If a straight line meets two straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.
An Axiom is a self-evident truth, and is so called from its being worthy (äčios) of universal acknowledgment. The above statement is not an Axiom, but the enunciation of a Theorem, easily capable of proof, but certainly requiring one. A proof is appended to the 29th Proposition.
N. B.—The “Elements" of Euclid are set forth in a series of Propositions, of which there are two forms, viz.:the Problem (z póßinua), and the Theorem (0ɛúpnua), indicating respectively a question proposed for resolution, and a fact submitted to demonstration. The results of the operation, in the former case, are marked by the letters Q. E. F. (quod erat faciendum), and in the latter, by Q. E. D. (quod erat demonstrandum.) Each Proposition consists of four parts:-1. The general enunciation, or statement of the object to be effected or proved. 2. The enunciation repeated with reference to a particular figure. 3. The construction, by means of certain lines drawn, or points determined, in order to complete the solution or proof. 4. The demonstration, either by a direct process of reasoning, or, indirectly, by proving a contrary supposition to be absurd.—The enunciation of a Problem proposes certain data, or premises, from whence to deduce a required result: that of a Theorem is divided into the hypothesis, or fact assumed to be true, and the conclusion built thereon, and requiring demonstration. A corollary is an easy inference from the whole or part of a preceding demonstration. . These several parts will be distinguished in each Proposition. In the General Enunci. ations, the data and hypotheses will be printed in Roman characters, the requisition or conclusion in Italics: in the Particular Enunciation this distinction will be indicated by