One of the alterations made with this view, respects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of EUCLID, has great advantages accompanied with considerable defects; of which, however, it must be observed, that the advantages are essential, and the defects only accidental. To explain the nature of the former requires a more minute examination than is suited to this place, and must therefore be reserved for the Notes; but, in the mean time, it may be remarked, that no definition, except that of EUCLID, has ever been given; from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. As to the defects, the prolixness and obscurity that have so often been complained of in the fifth Book, they seem to arise chiefly from the nature of the language employed, which being no other than that of ordinary discourse, cannot express, without much tediousness and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no assistance can be received from diagrams. As it is plain that the concise language of Algebra is directly calculated to remedy this inconvenience, I have endeavoured to introduce it here, in a very simple form however, and without changing the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this means, the steps of the reasoning which were before far separated, are brought near to one another, and the force of the whole is so clearly and directly perceived, that I am persuaded no more difficulty will be found in understanding the propositions of the fifth Book than those of any other of the Elements. In the second Book, also, some algebraic signs have been introduced, for the sake of representing more readily the addition and subtraction of the rectangles on which the demonstrations depend. The use of such symbolical writing, in translating from an original, where no symbols are used, cannot, I think, be regarded as an unwarrantable liberty: for, if by that means the translation is not made into English, it is made into that universal language so much sought after in all the sciences, but destined, it would seem, to be enjoyed only by the mathematical. The alterations above mentioned are the most material that have been attempted on the books of EUCLID. There are, however, a few others, which, though less considerable, it is hoped may in some degree facilitate the study of the Elements. Such are those made on the definitions in the first Book, and particularly on that of a straight line. A new axiom is also introduced in the room of the 12th, for the purpose of demonstrating more easily some of the properties of parallel lines. In the third Book, the remarks concerning the angles made by a straight line, and the circumference of a circle, are left out, as tending to perplex one who has advanced no farther than the elements of the science. Some propositions also have been added; but for a fuller detail concerning these changes, I must refer to the Notes, in which several of the more difficult, or more interesting subjects of Elementary Geometry are treated at considerable length. ELEMENTS OF GEOMETRY. BOOK I. THE PRINCIPLES. EXPLANATION OF TERMS AND SIGNS. 1. Geometry is a science which has for its object the measurement of magnitudes. Magnitudes may be considered under three dimensions,-length, breadth, height or thickness. 2. In Geometry there are several general terms or principles; such as, Definitions, Propositions, Axioms, Theorems, Problems, Lemmas, Scholiums, Corollaries, &c. 3. A Definition is the explication of any term or word in a science, showing the sense and meaning in which the term is employed. Every definition ought to be clear, and expressed in words that are common and perfectly well understood. 4. An Axiom, or Maxim, is a self-evident proposition, requiring no formal demonstration to prove the truth of it; but is received and assented to as soon as mentioned. Such as, the whole of any thing is greater than a part of it; or, the whole is equal to all its parts taken together; or, two quantities that are each of them equal to a third quantity, are equal to each other. 5. A Theorem is a demonstrative proposition; in which some property is asserted, and the truth of it required to be proved. Thus, when it is said that the sum of the three angles of any plane triangle is equal to two right angles, this is called a Theorem; and the method of collecting the several arguments and proofs, and laying them together in proper order, by means of which the truth of the proposition becomes evident, is called a Demonstration. 6. A Direct Demonstration is that which concludes with the direct and certain proof of the proposition in hand. It is also called Positive or Affirmative, and sometimes an Ostensive Domonstration, because it is most satisfactory to the mind. 7. An Indirect or Negative Demonstration is that which shows a proposition to be true, by proving that some absurdity would necessarily follow if the proposition advanced were false. This is sometimes called Reductio ad Absurdum; because it shows the absurdity and falsehood of all suppositions contrary to that contained in the proposition. 8. A Problem is a proposition or a question proposed, which requires a solution. As, to draw one line perpendicular to another; or to divide a line into two equal parts. 9. Solution of a problem is the resolution or answer given to it. A Numerical or Numeral solution, is the answer given in numbers. A Geometrical solution, is the answer given by the principles of Geometry. And a Mechanical solution, is one obtained by trials. 10. A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it 11. A Corollary, or Consectary, is a consequence drawn immediately from some proposition or other premises. 12. A Scholium is a remark or observation made on some foregoing proposition or premises. 13. An Hypothesis is a supposition assumed to be true, in order to argue from, or to found upon it the reasoning and demonstration of some proposition. 14. A Postulate, or Petition, is something required to be done, which is so easy and evident that no person will hesitate to allow it. 15. Method is the art of disposing a train of arguments in a proper order, to investigate the truth or falsity of a proposition, or to demonstrate it to others when it has been found out. This is either Analytical or Synthetical. 16. Analysis, or the Analytic method, is the art or mode of finding out the truth of a proposition, by first supposing the thing to be done, and then reasoning step by step, till we arrive at some known truth. This is also called the Method of Invention, or Resolution; and is that which is commonly used in Algebra. 17. Synthesis, or the Synthetic Method, is the searching out truth, by first laying down simple principles, and pursuing the consequences flowing from them till we arrive at the conclusion. This is also called the Method of Composition; and is that which is commonly used in Geometry. 18. The sign (or two parallel lines), is the sign of equality; thus, A=B, implies that the quantity denoted by A is equal to the quantity denoted by B, and is read A equal to B. 19. To signify that A is greater than B, the expression A7B is used. And to signify that A is less than B, the expression A/B is used. 20. The sign of Addition is an erect cross; thus A+B implies the sum of A and B, and is called A plus B. 21. Subtraction is denoted by a single line; as A-B, which is read A minus B; A-B represents their difference, or the part of A remaining, when a part equal to B has been taken away from it. In like manner, A-B+C, or A+C—B, signifies that A and C are to be added together, and that B is to be subtracted from their sum. 22. Multiplication is expressed by an oblique cross, by a point, or by simple apposition: thus, AXB, A. B, or AB, signifies that the quantity denoted by A is to be multiplied by the quantity denoted by B. The expression AB should not be employed when there is any danger of confounding it with that of the line AB, the distance between the points A and B. The multiplication of numbers cannot be expressed by simple apposition. 23. When any quantities are enclosed in a parenthesis, or have a line drawn over them, they are considered as one quantity with respect to other symbols: thus, the expression AX(B+C-D), or AXB+C-D, represents the product of A by the quantity B+C-D. In like manner, (A+B)×(A—B+C), indicates the product of A+B by the quantity A-B+C. 24. The Co-efficient of a quantity is the number prefixed to it: thus, 2AB signifies that the line AB is to be taken 2 times; AB signifies the half of the line AB. 25. Division, or the ratio of one quantity to another, is usually denoted by placing one of the two quantities over the other, in the form of a fraction: thus, A B signifies the ratio or quotient arising from the division of the quantity A by B. In fact, this is division indicated. 26. The Square, Cube, &c. of a quantity, are expressed by placing a small figure at the right hand of the quantity: thus, the square of the line AB is denoted by AB2, the cube of the line AB is designated by AB3; and so on. 27. The Roots of quantities are expressed by means of the radical sign√, with the proper index annexed; thus, the square root of 5 is indicated √5; √(AB) means the square root of the product of A and B, or the mean proportional between them. The roots of quantities are sometimes expressed by means of fractional indices: thus, the cube root of AXBXC may be expressed by AXBXC, or (AXBXC), and so on. 28. Numbers in a parenthesis, such as (15. 1.), refers back to the number of the proposition and the Book in which it has been announced or demonstrated. The expression (15. 1.) denotes the fifteenth proposition, first book, and so on. In like manner, (3. Ax.) designates the third axiom; (2. Post.) the second postulate; (Def. 3.) the third definition, and so on. To express either of 29. The word, therefore, or hence, frequently occurs. these words, the sign .. is generally used. 30. If the quotients of two pairs of numbers, or quantities, are equal, the quantities are said to be proportional: thus, if A C = B 'D D; then, A is to B as C to D. And the abbreviations of the proportion is, A: B:: C: D; it is sometimes written A: B⇒C: D. DEFINITIONS. 1. "A POINT is that which has position, but not magnitude*." (See Notes.) 2. A line is length without breadth. "COROLLARY. The extremities of a line are points; and the intersections "of one line with another are also points." 3. "If two lines are such that they cannot coincide in any two points, with"out coinciding altogether, each of them is called a straight line.” "COR. Hence two straight lines cannot inclose a space. Neither can two straight lines have a common segment; that is, they cannot coincide "in part, without coinciding altogether." 4. A superficies is that which has only length and breadth. "COR. The extremities of a superficies are lines; and the intersections of one superficies with another are also lines." 66 5. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies. 6. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. A D E B N. B. 'When several angles are at one point B, any one of them is ex'pressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and one of these 'two is somewhere upon one of those straight lines, and the other upon the 'other line: Thus the angle which is contained by the straight lines, AB, 'CB, is named the angle ABC, or CBA; that which is contained by AB, * The definitions marked with inverted commas are different from those of Euclid. |