A single capital letter, in reference to a diagram, as A, or B, denotes the point A, or the point B.t

Two capital letters, also in reference to a diagram, as A B, or CD, denote the straight line A B, or C D, or the side of a triangle, or other figure.

2

Two capital letters, with the figure just above to the right hand, as A B2, denote, not the square of A B, but the square on the straight line A B.

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

* When an s is added to the sign, the plural is denoted.

+N.B. A single capital letter may also denote an angle, when the sign, or word angle is used. Also, a parellelogram may be named by naming the single letters at the opposite angles.

Q.E.D., quod erat demonstrandum, which was the thing to be proved.
Q.E.F., quod erat faciendum, which was the thing to be done.

ad imposs. ad impossibile, reduced

to an impossibility.
a fortiori, by a stronger
argument or reason.

fig......figure.

....alternate.

...assumendo, by assuming,
adopting, or taking.

qu. lat..

......

..quadrilateral.

SECTION III.

EXPLANATION OF SOME GEOMETRICAL TERMS.

A Definition (from definire, to set bounds to) is a short description of a thing by such of its properties as serve to distinguish it from all other things of the same kind.

A Postulate (postulatum, a thing demanded) is a self-evident problem, the admission of which is demanded without formal proof.

An Axiom (axioma, a thing of worth) is a self-evident theorem, or the assertion of a truth, which does not need demonstration : it is worthy of credit as soon as stated.

A Proposition (proponere, to set forth) is something proposed to be done, as a problem; or to be proved, as a theorem.

A Problem (probleema, a thing proposed) is a proposal to do a thing, to construct a figure, or to solve a question.

A Theorem (theoreema, a subject of contemplation) is the assertion of a geometrical truth, and requires demonstration.

The Data (datum, a thing granted) are the things granted in a problem;

The Quæsita (quæsitum, a thing sought) are the things sought for in it;

The Hypothesis (hypothesis, a supposition) is the supposition made in a theorem ;

The Conclusion (concludere, to infer) is the consequence or inference deduced from it.

The General Enunciation (enunciare, to speak out, declare) of a proposition sets forth in general terms the conditions of the problem, or theorem, with what has to be done, or with what is inferred or concluded.

A Diagram (diagramma, a drawing of lines) is the drawing which represents a geometrical figure.

The Exposition (exponere, to set forth), or Particular Enunciation, sets forth the same conditions with an especial reference to a figure that has been drawn.

The Solution (solutio, an unloosening, an explaining) of a problem shows how the thing proposed may be done.

The Construction (constructio, a putting together) prepares, by the drawing of lines, &c., for the demonstration of a proposition.

The Demonstration (demonstrare, to point out) proves that the process indicated in the solution is sound, or that the conclusion deduced from an hypothesis is true-i.e., in accordance with geometrical principles.

The Recapitulation (recapitulare, to go over the main points again), or Conclusion, is simply the repetition of the proposition, or general enunciation, as a fact, or as a truth, with the declaration Q.E.F., or Q.E.D.

A Corollary (corolla, a little wreath, a deduction) is an inference made immediately from a proposition.

A Scholium (scholion, a comment) is a note or explanatory observation.

A Lemma (leema, a thing taken) is a preparatory proposition borrowed from another part of the same subject, and introduced for the purpose of establishing a more important proposition.

The Converse (conversum, a thing turned round) of a proposition is when the hypothesis of a former proposition becomes the conclusion, or predicate, of the latter proposition; as in Props. 5 and 6, 18 and 19, 21 and 25, bk. i.

The Contrary of a proposition is when that which the proposition assumes is denied.

Direct Demonstration is when the very thing asserted is proved to be true.

Indirect Demonstration is when all other cases, or conditions, except the one in question, are proved not to be true, and the inference is made: therefore, the very thing in question must be true; the assumption being that one out of several, or many, must be right.

The Position only of a line is meant, when the line is said to be given.

The Length only of a line is meant, when the line is said to be finite.

The Base of a figure (basis, a foundation) is the side on which it appears to stand; but each side, in turn, with the position of the figure changed, may become the base.

The Vertex (vertex, the top or crown of the head) is the highest angular point of a figure with a change of position in the figure, each angle may be named the vertical angle.

The Subtend (subtendere, to stretch under) of an angle is the side stretching across opposite to the angle.

The Hypotenuse (hupotenousa, that which stretches under) is the subtend to a right angle.

The Perpendicular (perpendiculum, a plumb-line) is the line forming with the base a right angle: lines are perpendicular to each other when at the point of junction they form a right angle.

B

A Figure is applied to a straight line when the line forms one of its boundaries.

The Altitude of a figure (altitudo, height) is the perpendicular distance from the side or angle opposite to the base, to the base itself, or to the base produced.

A Diagonal (diagonios, from corner to corner) is a straight line joining two opposite angular points.

The Complement of an angle (complementum, that which fills up) is what is wanted to make an acute angle equal to a right angle, or to 90°.

The Supplement of an angle (supplementum, a filling up) is what is wanted to make an angle equal to two right angles, or to 180°. The Explement of an angle (explementum, a filling) is what is wanted to make an angle equal to four right angles, or to 360°.

The Complements of a Parallelogram (see Fig. to Definition A.), when the parallelogram is bisected by its diagonal, and subsidiary parallelograms are formed by two lines-one, parallel to one side, and the other, parallel to the other side, and both intersecting the diagonal, the complements of the parallelogram are those subsidiary parallelograms through which the diagonal does not pass; and these, with the subsidiary parallelograms through which the diagonal does pass, fill up or complete the whole parallelogram.

The Area of a figure (area, an open space) is the quantity of surface contained in it, reckoned in square units, as square inches, square feet, &c.

A locus (locus, place) in Plane Geometry is a straight line, or a plane curve, every point of which, and none else, satisfies a certain condition.

SECTION IV.

NATURE OF GEOMETRICAL REASONING.

In Mechanics, Chemistry, and the kindred sciences we are so accustomed to experimental proof, and find it so very useful, that we readily suppose that experiment may be applied to Geometry equally well. Thus we draw a triangle, and at some