But as he supposes it proved not univerfally true, he presents us with prop. A, B, C, after prop. 23. Book XI. to fupply its defect. prop. C. "Solid figures contained by the fame num"ber of equal and fimilar planes alike fituated, and having none "of their folid angles contained by more than three plane angles, are equal and fimilar to one another." But this prop. C. will evidently appear infufficient to supply this fuppofed defect, on account of the limited fenfe in which it is taken; for, if folid figures, bounded by an equal number of equal and fimilar planes, are not equal and fimilar, but under this limitation, then prop. 15. Book V. must not be univerfally true, which I fuppofe will ́not eafily be admitted; and, if not admitted, then prop. C must be a very infufficient foundation for proof of the following propofitions depending on it, viz. Prop. 25. 26. and 28. and confequently eight others, viz. 27th, 31ft, 32d, 33d, 34th, 36th, 37th, and 4cth. Book XI. all which are by this author toffed off their bafe, which is univerfally true, and placed upon this limited one. Mr Simpson farther objects, that though this definition be true, yet ought not to be a definition, but a propofition, and the truth of it proved. The fame objection might be made with equal propriety to feveral others; for example, why not prove the equality of these angles which determine the equal inclination of planes, Def. 7. Book XI. and the equality of right lines equally diftant from the center, both which we may conclude to be Euclid's, as Mr Simpfon does not object to them; for he would make us believe none are Euclid's that he does not affirm to be fo, and that frequently without any other reafon given for it, but his own ipfe dixit. If we confider the nature of a definition, it is, if I miftake not, diftinguishing bodies from one another, by fuch properties as cannot be applied to any other bodies, but thofe it is intended to distinguish. In which fenfe, if the properties given in this definition are fuch as diftinguifh fimilar and equal bodies from others that are not so in every inftance, then it is certainly a proper definition; but Euclid has fometimes thought proper to prove his definitions; for example, def. 4. Book III. which he has proved, prop. 14. of that book. This, it would appear, he has not thought neceffary to prove, probably, if we may be allowed to affign a reafon in his name, that he has thought it so self-evident, that none would ever call the truth of it in queflion; but as the truth of it has been called in queftion, the definition may be proved in the following manner from Mr Simpfon's demonstration to prove the contrary; for which obferve his own figure and demonftration. He has proved the three triangles EAB, EBC, ECA, containing the one folid, equal and fimilar to the three triangles FAB, FBC, FCA, containing the other fo lid lid, having the common base, ABC; then, if the folid EABC is not equal to the folid FABC, let it be equal to fome folid as GA BC, either greater or less than EABC, which cannot be; for the one would contain the other; and if the folid angle is con tained by more than three plane angles, equal and fimilar to one another, then it can be divided into angles which are contained by three equal and fimilar plane angles, by Prop. 20. Book VI. and parts have the fame proportion as C their like multiples, by Prop 15. Book V. wherefore universally, figures bounded by an equal number of equal and fimilar planes are equal and fimilar. N. B. In the references, when the propofition referred to is in the fame book with the propofition to be proved, the book is not named, but only the number of the propofition, but, if in other book, both are named. any A right line is that which lieth evenly between its points. V. A fuperficies is that which hath only length and breadth. VI The bounds of a fuperficies are lines. VII. A plain fuperficies is that which lieth evenly between its lines. VIII. A plain angle is the inclination of two lines to one another in the fame plain, which touch each other, but do not lie in the fame right line. IX. If the lines containing the angle be right ones, then the angle is called a right-lined angle. X. When one right line ftanding on another right line makes the angles on each fide thereof equal to one another, each of thefe angles is a right one, and that line which ftands upon the other is called a perpendicular to that whereon it stands. A XI. An BOOK I. An obtufe angle is that which is greater than a right one. XII. An acute angle is that which is less than a right one. XIII. A term, or bound, is the extreme of any thing. XIV. A figure is that which is contained under one or more terms. XV. A circle is a plain figure bounded by one line, called the circumference, to which all right lines drawn from a certain point within the fame are equal. The diameter of a circle is a right line drawn through the center, and terminated on both ends by the circumference, and divides the circle into two equal parts. XVIII. A femicircle is a figure contained under any diameter, and the circumference cut off by that diameter. XIX. A segment of a circle is a figure contained under a right line, and circumference cut off by that right line. XX. Right-lined figures are fuch as are contained by right lines. Three fided figures are fuch as are contained by three lines. Four sided figures are fuch as are contained by four lines. XXIII. Many fided figures are fuch as are contained by more than four lines. XXIV. An equilateral triangle is that which hath three equal fides. XXV. An ifofceles triangle, that which hath two fides equal. XXVI. A fcalene triangle, that which hath all the three fides unequal. XXVII. A right angled triangle is that which hath one right angle in it. An obtufe angled one, that which hath one obtufe angle in it. XXIX. An acute angled triangle is that which hath all the angles lefs than right ones. XXX. A XXX. A fquare is that which hath four equal fides, and its angles all right ones. XXXI. An oblong, or rectangle, is longer than broad, its oppofite fides are equal, and its angles all right ones. XXXII. A rhombus, that which hath four equal fides, but not right angles. XXXIII. A rhomboides, whofe oppofite fides and angles are equal. XXXIV. All quadrilateral figures beside these are called trapezia. XXXV. Parallel right lines are fuch as, being produced both ways in the fame plain, never meet. XXXVI. A parallelogram is a figure whofe oppofite fides are parallel. G POSTULATE S. I. RANT that a right line may be drawn from any one II. That a finite right line may be continued directly forwards: And, III. That a circle may be defcribed about any center, with any distance. T HINGS equal to one and the fame thing are equal to II. If equal things are added to equal things, the wholes will be equal. III. If from equal things equal things be taken, the remainders will be equal. IV1 If to unequal things equal things are added, the whole will be unequal, V. I£ Book I. |