GH and at the fame time greater, as that it fhould be lefs than GH and at the fame time greater. There is fomething rather fingular in the eighteenth propofition, for it seems to me to be nothing but the corollary to the fixteenth propofition; because a tangent there; is a straight line at right angles upon the extremity of the diameter; according this notion therefore the one proves that AE is perpendicular to AB; and the other that AB is perpendicular to AE which is certainly the fame thing. Unless it should be urged in favour of Euclid that the corollary does not fay that every tangent is at right angles to the diameter paffing through the point of contact, but only if it be at right angles it will be a tangent; but it appears to me that there is no room for this diftinction in the present case; for when we say that AE is perpendicular to AB; it is not the converfe of that propofition to say that AB is perpendicular to AE. It would be a proper exercife for the student in reading the feventeenth propofition to prove that two tangents might be drawn from any point without a circle, and that they are equal, though this is not according to Euclid's plan.

I come now to the twentieth propofition, which should be examined with the greatest care, by describing figures for every distinct cafe, and varying the pofition of the angular point through the whole circumference BADC: after this is done it will be proper for the student to make an obtufe angle at the circumference, which will give him a particular kind of angle at the center, and it might admit of fome difpute whether it is to be reckoned an angle in Euclid's fenfe of the word; however it may be proved to be double of the obtufe angle at the circumference of the circle. As this is a very fundamental propofition, the reader may just observe, (what I have recommended to him for a conftant practice) the confequences which are drawn from every part of the fuppofition; the angles are faid to be at the center and circumference; and to stand. upon the fame circumference. Now if either of thefe circumftances be omitted, the triangles will not be ifofceles &c. The next propofition fhould alfo be particularly examined in all its cafes; when the fegment is greater than a femicircle, equal to a femicircle, and lefs than one; indeed if an angle increafing until the lines take the fame direction is to be confidered as equal to two

right angles, and if what an angle wants of four right angles is to be regarded as an angle, there is but one cafe of the propofition; however if these do not make an angle, yet by drawing a line through the center they will be angles in Euclid's fenfe of the term, and the demonftration of the other two cafes will be obvious.

I shall conclude my remarks upon this book by defiring the reader to take notice of a theorem which follows from the thirty third propofition; viz. If two triangles have their bafes equal, and alfo the angles under which the bases are extended; and if they be between the fame parallel lines; the two triangles will be equal in every refpect. This is mentioned to fhew that the joining the properties of the circle to those of the triangle will enlarge our notions very much; because we never could have discovered this by the triangle alone.

In the fifth propofition of the fourth book, Simfon thinks that Euclid should have proved that the perpendiculars to the fides of the triangles will meet; and it seems to me equally neceffary to fhew that the tangents will meet in the third propofition; indeed if they did not the angle AKB would cease to be an angle, and AKB would be a straight, but it is equal to the outward angle of a triangle therefore &c. Or in both these instances it may be demonstrated as Clavius has done by joining AB in the one; and DE in the other; which brings them to the eleventh common notion.

But although I recommend it to every ftudent to examine all poffible fuppofitions, yet for reafons which have been mentioned already fuch omiffions seem to me very confiftent with Euclid's plan of demonstration; for he supposes his reader attentive; and therefore gives him only, what he judged to be the necessary affis

tance.

THE

ELEMENTS

O F

EUCLI D.

BOOK I.

I.

A

DEFINITION S.

POINT is that of which no part can be taken.

2. But A LINE bas length without breadth. 3. Alfo extremities of a line are points. 4. Any line which lies evenly between the points in itself is A STRAIGHT LINE. 5. And that is A SURFACE which has length and breadth only. 6. Also extremities of a surface, are lines. 7. Any furface which lies evenly between the straight lines in itself is a PLANE. 8. But A PLANE ANGLE is the inclination of lines to one another; that is of two lines in a plane, meeting each other and not lying in a straight line. 9. Alfo it is called A RECTILINEAL ANGLE, when the lines containing the angle are straight. 10. But when a straight line standing upon a straight line makes the adjacent angles equal to one another; each of the equal angles is A RIGHT ANGLE: and the ftanding straight line is called A PERPENDICULAR to that on VOL. I. which

Α

Book I. which it ftands.

a FIGURE.

II. AN OBTUSE ANGLE is that which is greater than a right angle. 12. But AN ACUTE angle is that which is lefs than a right angle. 13. That which is an extremity of any thing is called A TERM. 14. The space bounded by one or more terms is called 15. A CIRCLE is a plane figure bounded by one line, which is called a CIRCUMFERENCE: upon which all the straight lines falling, from ONE POINT of those lying within the figure, are equal to one another. 16. And the point is called the CENTER of the circle. 17. But a DIAMETER of a circle is any right line drawn through the center, and terminated both ways by the circumference of the circle: which alfo divides the circle into two equal parts. 18. Alfo A SEMI-CIRCLE is the figure bounded by a diameter, and the circumference of the circle cut off by it. 19. A SEGMENT of a circle is the figure bounded by any straight line and the circumference of a circle. 20. The figures bounded by straight lines are called RECTILINEAL FIGURES. 21. Of thefe, fuch as fuch as are bounded by three, are called TRILATERAL. 22. As those by four QUADRILATERAL. 23. But those bounded by more than four straight lines are called MULTILATERAL. 24. Again of trilateral figures, AN EQUILATERAL TRIANGLE particularly is that which has three equal fides. 25. But an ISOSCELES, that which has only the two fides equal. 26. And a SCALENE, that which has the three fides unequal. 27. But moreover of trilateral figures, that again is a RIGHT-ANGLED TRIANGLE, which has a right angle. 28. And an OBTUSE ANGLED triangle, that which has an obtufe angle. 29. Also that which has three acute angles, an ACUTE-ANGLED triangle. 30. But of quadrilateral figures, that is a SQUARE, which is equilateral and rectangular. 31. And an OBLONG, that which though rectangular is not equilateral. 32. And a RHOMBUS, though equilateral, is not rectangular. 33. But that which is neither equilateral, nor rectangular, having only its oppofite fides and angles equal to one another, is a RHOMBOIDES. 34. And all other quadrilateral figures except these may be called TRAPEZIUMS. 35. Any ftraight lines, which are in the fame plane, and being produced indefinitely towards both fides meet each other on neither, are PARALLELS.

POSTULATE S.

1. Let it be taken for granted, that a ftraight line may be drawn from any one point to any other point. 2. And that a finite straight line may be produced in a straight line continually. 3. Also that a circle may be described with any center and at any distance.

COMMON NOTION S.

1. Magnitudes, which are equal to the fame magnitude, are equal to one another. 2. And if equal ones be added to equal ones, the wholes are equal. 3. And if from equals, equals be taken away, the remainders are equal. 4. And if to unequals, equals be added, the wholes are unequal. 5. Also if from unequals, equals be taken away, the remainders are unequal. 6. And the doubles of the fame are equal to one another 7. Alfo the halves of the fame are equal to one another. 8. Magnitudes which fit each other exactly are equal to one another. the whole is greater than its part. 10. And all right angles are equal to one another. II. And if a straight line meeting two ftraight lines, make those angles, which are inward and upon the fame fide of it, less than two right angles, the two straight lines being produced indefinitely will meet each other on the fide where the angles are less than two right angles. 12. Also two ftraight

lines do not inclose a space.

PROPOSITION I

9. Alfo

Upon a given finite straight line to describe an equilateral triangle.

Let AB be the given finite straight line: it is required to defcribe an equilateral triangle upon the ftraight line AB.

With the center A and at the distance AB (by poft. 3.) let the circle BCD be described: and again, with the center B, but at

Book I.