PART I.

GRADATIONS IN EUCLID.

ELEMENTS OF PLANE GEOMETRY.

BOOKS I. & II.

In using the letters that have reference to the Figures or Diagrams, let it be remembered:

1o. That a single capital letter, as A, or B, denotes the point A, or the point B.

A single capital letter may also denote an angle, when the sign or the word angle is used with it;

And a parallelogram may be named by naming the letters at the opposite angles; as BK and C K, in the diagram on p. 17.

2o. Two capital letters, as A B, or CD, denote the st. line AB, or CD, or the side of a triangle or other st. lined figure.

2

But two capital letters, having the numeral just above to the right hand, as A B2, denote, not the square of st. line A B, but the square on st. line AB.

3o Capital letters with a point between them, as AB. CD, denote, not the product of AB multiplied by CD, but the rectangle formed by two of its sides meeting in a common point.

GRADATIONS IN EUCLID.

BOOK I.

THE GEOMETRY OF PLANE TRIANGLES.

"IN THIS FIRST BOOKE," says Sir Henry Billingsley, who in 1576 published the earliest translation of Euclid into English, "is intreated of the most simple, sure, and first matters and groundes of Geometry; as, namely, Lynes, Angles, Triangles, Parallels, Squares, and Parallelogrammes. First of theyr definitions, shewyng what they are. After that it teacheth how to draw Parallel lynes, and how to forme diuersly figures of three sides, and foure sides, according to the varietie of their sides and Angles; and compareth them all with Triangles, and also together the one with the other. In it also is taught how a figure of any forme may be chaunged into a figure of an other forme. And for that it entreateth of these most common and general thynges, thys booke is more vniuersall than is the seconde, third, or any other, and therefore iustly occupieth the first place in order; as that without which the other bookes of Euclide which follow, and also the workes of others which haue written on Geometry, cannot be perceaued nor vnderstanded. And forasmuch as all the demonstrations and proofes of all the propositions in this whole booke, depende of these groundes and principles following, which by reason of their playnnes néede no greater declaration, yet to remoue all (be it neuer so litle) obscuritie, there are here set certayne short and manifest expositions of them."-Billingsley's Euclid, a.D. 1570.

DEFINITIONS.

1. A Point (punctum, a small hole) is that which has no parts, or which has no magnitude: it marks position. PYTHAGORAS.

A point is that of which there is no part."-EUCLID: or,
be parted or divided."-PROCLUS.

THEON and

"6 which cannot

A mathematical

"A point is a monad having position."-PYTHAGORAS.
point cannot be drawn ; for a visible point is, in fact, a surface.

2. A Line (linea, a linen thread) is length without breadth; or it is extension in one direction.

A mathematical line cannot be drawn; for whatever is visible must have breadth.

A line is measured by the number of units, or monads, of length contained in it; as, 5 inches; 9 feet; 13 miles.

3. The extremities (extremus, outermost), or ends, of a line are points.

4. A right, or straight line, is that which lies evenly between its extreme points.

"A straight line is the shortest distance between two points."-ARCHIMEDES, adopted by LEGENDRE.

"A straight line is that of which the extremity hides all the rest, the eye being placed in the continuation of the line."-PLATO. Plato's line was thus a visible, not a mathematical line.

5. A superficies (super, over; facies, a face), or surface, is that which has only length and breadth: it is extension in two directions. Such a surface is merely the outside, without thickness.

any

6. The extremities, or boundaries, of a surface, are lines.

7. A plane surface is that in which any two points being taken, the straight line joining them lies wholly in that surface.-HERŌN THE ELDER.

"A plane surface is that which lies evenly, or equally, with the straight lines in it."-EUCLID.

"A plane surface is one whose extremities hide all the intermediate parts, the eye being placed in its continuation."-PLATO. "A plane surface is the smallest surface which can be contained between given extremities." "A plane surface is that to which a straight line may be applied in any manner of way."