2. That a terminated straight line may be produced to any length in a straight line: 3. That a circle may be described from any centre, at any distance from that centre.* AXIOMS. 1. Things which are equal to the same, or to equals, are equal to one another. 2. If equals or the same be added to equals, the wholes are equal. 3. If equals or the same be taken from equals, the remainders are equal. 4. If equals or the same be added to unequals, the wholes are unequal. 5. If equals or the same be taken from unequals, the remainders are unequal.+ 6. Things which are doubles of the same, or of equals, are equal to one another. 7. Things which are halves of the same, or of equals, are equal to one another.‡ 8. Magnitudes which exactly coincide with one another, are equal. 9. The whole is greater than its part.§ 10. The whole is equal to all its parts taken together.|| 11. All right angles are equal to one another. 12. If a straight line meet two other straight lines which are in * These postulates require in substance that the simplest cases of drawing straight lines, and describing circles be admitted to be known; and they imply the use of the rule and compasses, or something equivalent. A circle may also be described by means of a cord or other line of invariable length, fixed at one extremity, or by employing another circle, already drawn, as a pattern. The latter mode, however, does not agree with the third postulate, as it does not enable us to describe a circle from a given centre, and at a given distance from that centre. In this axiom and the foregoing, it is evident that the result obtained by adding to the greater or taking from it, is greater than that which is obtained by adding to the less, or taking from it. Both this axiom and the preceding might be derived from the first. They are generalized in the axioms of the fifth book. § This is implied in the significations of the terms whole and part. In strictness, therefore, it is scarcely to be regarded as an axiom. Though this axiom is not delivered by Euclid in a distinct form, it is tacitly employed by him in many instances. Like the foregoing, it may be regarded as being implied in the signification of the terms whole and part. This axiom does not relate to the right angles made by one line standing on another, such angles being equal by the eighth definition. Instead of being considered as an axiom, this proposition has been demonstrated by some writers in the following manner: the same plane, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines shall at length meet upon that side, if they be continually produced.* PROPOSITION I. PROBLEM.+ To describe an equilateral triangle on a given finite straight line.‡ Let AB be the given straight line; it is required to describe an equilateral triangle upon it. Let AB be perpendicular to CD, and EF to GH. then the angles ABD, EFH are equal. For let the A B D F K F H straight line CD, be applied to GH, * This will be illustrated in the remarks on the 28th proposition of the first .. book; and the student may postpone the consideration of it, till he has proved that proposition. † The words in which a proposition is expressed, are called its enunciation. If the enunciation respect a particular diagram, it is called a particular enunciation otherwise, it is a general one. A demonstration is a series of arguments which establish the truth of a theorem, or of the solution of a problem. Demonstrations are either direct or indirect. The direct demonstration commences with what has been already admitted, or proved, to be true, and from this deduces a series of other truths, each depending on what precedes, till it finally arrives at the truth to be proved. In the indirect or negative demonstration, or as it is also called, the reductio ad absurdum, a supposition is made which is contrary to the conclusion to be established. On this assumption, a demonstration is founded, which leads to a result contrary to some known truth; thus proving the truth of the proposition, by showing that the supposition of its contrary leads to an absurd conclusion. The drawing of any lines, or the performing of any other operation that may be necessary in a proposition, is called its construction. In this proposition, the first paragraph is the general enunciation; the second, the particular one; the third, the construction; and the fourth, the demonstration. That is, to describe an equilateral triangle, which shall have a given straight line as one of its sides. The word finite is employed to show that the line is not of unlimited length, but is given in magnitude, as well as position. The student should be accustomed to point out the data in problems and the hypotheses in theorems. In this problem, a straight line is given, and it is required to describe on it an equilateral triangle. From the centre A, at the distance AB, describe (I. postulate 3.*) the circle BCD, and from the centre B, at the distance BA, describe (I. post. 3.) the circle ACE; and from the point C, in which the circles cut one another, draw (I. post. 1.) the straight lines CA, CB to the points A, B: ABC is the equilateral triangle required. E Because the point A is the centre of the circle BCD, AC is equal (I. definition 30.) to AB; and because the point B is the centre of the circle ACE, BC is equal (I. def. 30.) to BA. But it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB: but things which are equal to the same are equal (I. axiom 1.) to one another; therefore CA is equal to CB; wherefore CA, AB, BC are equal to one another; and the triangle ABC is therefore (I. def. 17.) equilateral, and it is described upon the given straight line AB: which was required to be done. Scholium. If straight lines be drawn from A and B, to F, the other point in which the circles cut one another, it would be proved in the same manner that AFB is an equilateral triangle. Hence on any straight line two equilateral triangles may be described one on each side of it.‡ PROP. II. PROB. FROM a given point to draw a straight line equal to a given straight line. § Let A be the given point, and BC the given straight line; it is required to draw from A a straight line equal to BC. From the point A to B draw (I. post. 1.) the straight line AB; and upon it describe (I. 1.) the equilateral triangle DAB, and produce (I. post. 2.) the straight lines DB, DA, to E and F. From the centre B, at the distance BC, describe (I. post. 3.) the circle CEH, and from the G H D E A * In the references, the Roman numerals denote the book, and the others, when no word is annexed to them, indicate the proposition ;-otherwise the latter denote a definition, postulate, or axiom, as specified. Thus, III. 16. means the sixteenth proposition of the third book; and I. ax. 2., the second axiom of the first book. So also hyp. denotes hypothesis, and const. construction. By a scholium is meant a note or observation. It will be shown in the 10th proposition of the 3d book, that two circles can cut each other in only two points; and hence there can be only two equilateral triangles on a straight line. In the practical construction it is sufficient to describe small arcs intersecting each other in C or F. § Here the data are a point and a straight line. centre D, at the distance DE, describe (I. post. 3.) the circle EFG. AF is equal to BC.* Because the point B is the centre of the circle CEH, BC is equal (I. def. 30.) to BE; and because D is the centre of the circle EFG, DF is equal (I. def. 30.) to DE; and DA, DB, parts of them are equal: therefore the remainder AF is equal (I. ax. 3.) to the remainder BE. But it has been shown, that BC is equal to BE; wherefore AF and BC are each of them equal to BE; and things that are equal to the same are equal (I. ax. 1.) to one another; therefore the straight line AF is equal to BC. Wherefore from the given point A a straight line AF has been drawn equal to the given straight line BC: which was to be done. PROP. III. PROB. FROM the greater of two given straight lines to cut off a part equal to the less.† Let AB and C be the two given straight lines, of which AB is the greater. It is required to cut off from AB, the greater, a part equal to C, the less. From the point A draw (I. 2.) the straight line! AD equal to C; and from the centre A, at the distance AD, describe (I. post 3.) the circle DEF: AE is the part required. D A /E B Because A is the centre of the circle DEF, AE is equal (I. def. 30.) to AD; but the straight line C is likewise equal (const.) to AD; whence AE and C are each of them equal to AD; wherefore the straight line AE is equal (I. ax 1.) to C, and from AB, the greater of two straight lines, a part AE has been cut off equal to C the less: which was to be done. A straight line may be drawn from the point A, to either extremity of BC, and on either of the lines thus drawn, two equilateral triangles may be constructed. Hence there may be four straight lines drawn, any one of which will be such as is required in the problem. To this there is an exception, when A is at either extremity of BC, or in its continuation, as in either case it will readily appear that only two such lines, can be drawn by the construction here given. It is plain also that if A be in BC, the equilateral triangle is to be described on either of the parts into which BC is divided at that point. In practice, every person will solve this problem by opening the compasses to the distance between B and C ; and then, one point being placed at A, the other will mark out the point F on any line drawn from A. This method, however, though greatly preferable in practice, could not in strictness be adopted by Euclid, as it is not derived from the postulates or the first proposition. It is in fact, the same as assuming this proposition as a postulate; and in a science strictly demonstrative, the fewer first principles that are assumed the better; and though it may sometimes be convenient, and may save time, to dispense with such rigour, it is always satisfactory to know that the object in view may he attained without any departure from strict geometrical accuracy. Here the data are two straight lines. Schol. By drawing (I. 2.) from either extremity of C, a straight line equal to AB, the line C might be produced, till it and the part added would be equal to AB. PROP. IV. THEOREM. If two triangles have two sides of the one equal to two sides of the other, each to each ;† and have also the angles contained by those sides equal to one another : (1.) they have likewise their bases, or third sides,‡ equal; (2) the two triangles are equal;§ and (3.) their other angles are equal, each to each, viz., those to which the equal sides are opposite.|| A D Let ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides, DE, DF, each to each, viz., AB to DE, and AC to DF; and the angle BAC equal to the angle EDF: then (1.) the base BC is equal to the base EF; (2.) the triangle ABC to the triangle DEF; and (3.) the other angles, to which the equal sides are opposite, are equal, each to each, viz., the angle ABC to the angle DEF, and the angle ACB to DFE. B E C P H For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon. DE; the point In practice, what is done in this proposition and in the scholium, will be effected simply by means of the compasses, as was pointed out in the preceding proposition. †The meaning of the expression each to each, or respectively, which is used in the same sense, will be known from its application here. Were this expres sion wanting, the meaning might be merely that the sides AB, AC are together equal to DE, DF, while, when taken separately, they might be either equal or unequal. With the limiting expression, however, the meaning is, that AB is equal to DE, and AC to DF. In such cases, the lines or magnitudes must be taken in the same order. Thus, it would be improper in the present case to say, that AB, AC are equal to DF, DE, each to each. This expression shows the meaning which Euclid attaches to the base of a triangle. It is the third side as distinguished from the other two, whether that is the side on which the triangle stands or not. § That is, they have equal areas or surfaces, the area of a superficial figure being the space which it contains. This enunciation might be more briefly expressed thus: If two triangles have two sides and the contained angle of the one respectively equal to two sides and the contained angle of the other; they have likewise their remaining sides and angles equal, each to each, viz., those which are similarly situated; and their areas are equal. In this proposition, the hypothesis is, that there are two triangles which have two sides and the contained angle in one of them, equal respectively to two sides and the contained angle in the other; and it is proved that if this be so, the triangles must be in every respect equal. |