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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 (from postulatum, a thing demanded), is a self-evident problem, the admission of which is demanded without formal proof.
An Axiom (from 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 (from proponere, to put forth), is something proposed to be done, as a problem; or to be proved, as a theorem.
A Problem (from probleema, a thing proposed), is a proposal to do a thing, to construct a figure, or to solve a question.
A Theorem (from theoreema, a subject of contemplation), is the assertion of a geometrical truth, and requires demonstration.
The Data (from datum, a thing granted), are the things granted in a problem;
The Quæsita (from quæsitum, a thing sought), are the things sought for in it.
The Hypothesis (from hupothesis, a supposition), is the supposition made in a theorem;
The Conclusion (from concludere, to infer), is the consequence or inference deduced from it.
The General Enunciation (from enunciare, to speak out, or 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.
The Exposition (from exponere, to set forth), or Particular Enunciation, sets forth the same conditions with an especial reference to a diagram that has been drawn.
A Diagram (from diagramma, a drawing of lines), is the drawing which represents a geometrical figure.
The Solution (from solutio, an unloosening, an explaining), of a problem shows how the thing proposed may be done.
The Construction (from constructio, a putting together), prepares, by the drawing of lines, &c., for the demonstration of a proposition.
The Demonstration (from 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, (from recapitulare, to go over the main points again), 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 (leemma, 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 P. 5 and 6; 18 and 19 ; 24 and 25; book 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
Indirect Demonstration is when all other cases, or conditions, except the one in question, are proved not to be true, and the inference is madetherefore 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 (basis, a foundation), of a figure 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 (verter, 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.
A Figure is applied to a straight line when the line forms one of its
The Altitude (altitudo, height), of a figure 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 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°.
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 in Plane Geometry is a straight line, or a plane curve, every point of which, and none else, satisfies a certain condition.
NATURE OF GEOMETRICAL REASONING.
"Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning."-Locke, vol. 4, p. 428.
The Demonstrations in Euclid's Elements of Geometry consist of arguments by which the assertions made in the propositions are proved to be true. Thus, in the 15th Prop., bk. i., the assertion is made that "the opposite, or vertical angles, formed by two intersecting lines, are equal;" and the demonstration shows by argument, founded upon truths already admitted or proved, that the assertion itself must be received as true.
When fully stated, each argument contains both the thing which is proved, and the means by which the proof is established: the means of proof, usually preceding the thing proved, are named the premisses; and the thing proved is named the conclusion or inference. Thus, in Prop. 1, bk. i., the premisses are-1st, things equal to the same thing are equal to each other; 2nd, the line AC, and also the line BC, are each equal to the same line AB; and 3rd, the inference, or thing proved, is, that the line AC equals the line BC, i.e., the two lines AC and BC are equal to each other.
Here in the premisses two things are laid down, or granted to be true: as," things equal to the same thing are equal to each other,"-this is one truth; "the line AC equals the line AB, and the line BC also equals the same line AB,"-this is another truth; and from the two things thus declared to be true, there is made the unavoidable inference, therefore the two lines AC, BC, are equal to each other.
In this mode of reasoning, it is seen that assertions are broadly made; and we may ask, on what evidence are they to be received as true?
The first kind of evidence is from the definition of the thing; thus, we define a triangle to be a figure bounded by three sides; and if, of any figure placed before us, we can affirm, that it has three sides exactly, the conclusion is inevitable, that this figure also is a triangle.
The second kind of evidence is from the axioms, i.e. from truths so plain that they need no proof: for example,- -we receive as undeniable, that, if equals be added to equals, the wholes are equal; and we argue, if to the line AD, or to its equal the line BC, we add another line EF, then the whole line made up of AD+EF, will equal the whole line made up of BC + EF.
The third kind of evidence is from the hypothesis, or from the supposition, which we make as the condition of our assertion: we declare, "in an isosceles triangle, the angles at the base are equal;" the very words, though not in the exact form of an hypothesis, directly imply the supposition, " if a triangle is isosceles," "then the angles at the base are equal." An isosceles triangle is here taken as the starting point of the reasoning;-and though, for the demonstration of the inference, "the angles at the base are equal," it is necessary to draw various lines which are not mentioned in the hypothesis, the conclusion at which we arrive is altogether dependent on the hypothesis.
The fourth kind of evidence is from proof already given; for what has once been established, may afterwards be taken for granted. For instance, when we have once established the truth, that" the interior angles of every triangle are together equal to two right angles," and we afterwards come to a proposition in the demonstration of which we need this established truth, we do not again go through all the steps by which the equality of the sum of the interior angles of a triangle to two right angles has been proved, but, we make use of that truth as unquestionable, and argue directly from it.
But the Principle of Geometrical Reasoning is, that from two propositions established or received as true, a third proposition or inference shall be made. Now, that this may be done, there must be something in common contained in both the propositions, with which common thing, as with a test or standard, the other two things are compared: we say―
All the triangle is in the circle,
All the square is in the triangle,
therefore, All the square is in the circle :
the test, the common term of comparison, here is, "the triangle," and our inference is correct.
When a connexion is thus declared to exist between the premisses and the conclusion, that is, when reasons are stated and an inference made,this mode of argument receives the name of a Syllogism; (from sullogismos, a collection), for a Syllogism is a bringing together into one view the two steps of the reasoning on which a truth depends, and the truth itself; or, as Whately in his Elements of Logic, p. 52, defines a Syllogism, it is "an argument so expressed, that the conclusiveness of it is manifest from the mere force of the expression, i.e., without considering the meaning of the terms, e.g., in this Syllogism."
"the Conclusion is inevitable, whatever terms, X, Y, and Z respectively, are understood to stand for. And to this form all legitimate Arguments may ultimately be brought."
For further information the learner may consult the Author's, Plane Geometry Practically applied Part I., p. 9-17; or De Morgan's Study of Mathematics, p. 68-76.
INDUCTION is a species of argument in which that is inferred respecting a whole class, which has been ascertained respecting several individuals of the class: carried out completely, Inductive reasoning is that in which a universal proposition is proved by proving separately each of its particular cases: thus, the s, A, B, C, and D are all the s, in a certain figure ABCD; we prove that A is a rt. ▲, B a rt., Ca rt., and Dart. ; and we say, therefore all the angles of the figure ABCD are right angles. The argument a fortiori,' by the stronger reason, proves that a given predicate belongs in a greater degree to one subject than to another: as, A is greater than B; B greater than C; much more, a fortiori, is A greater than C. An example of this kind of argument occurs in Prop. 21, bk i. ;
The angle BDC is greater than the angle CED;
and angle CED is greater than the angle BAC;
much more.. is angle BDC greater than angle BAC.
The reductio ad impossibile,' the reduction to an impossibility, is when the argument shows that any given assertion is impossible; as in the 14th Prop., bk. i., where it is supposed that both the line BE and the line BD are continuations of another line CB; the demonstration conducts to the conclusion that the less angle ABE equals the greater angle ABD; but this is an impossibility, for the less cannot equal the greater.
There is also the reductio ad absurdum,' the reduction of an argument to an absurdity: it tak it takes place when the conclusion involves something foolish, or utterly unreasonable: thus, in Prop. 7, bk. i., the angle BDC is proved to be, first, equal to the angle BCD, and next, greater than the same angle BCD; but this is an absurdity. There is little real difference in Geometry between the impossible and the absurd.
THE GEOMETRY OF PLANE TRIANGLES.
"IN THIS FIRST BOOKE," says Sir Henry Billingsley, who in 1570 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 then 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 neede 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.
1. A Point (punctum, a small hole), is that which has no parts, or which has no magnitude: it marks position.-THEON and PYTHAGORAS. "A point is that of which there is no part."-EUCLID.
A mathematical point cannot be drawn ; for a visible point is, in fact, a surface. 2 2. A Line (linea, a linen thread), is length without breadth; or 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 of length contained in it;-as, 5 inches; 9 feet; 13 miles..
3. The Extremities (extremus, outermost), of a line are points.
5. A superficies, (super, above, 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 any thickness.
6. The extremities of a superficies are lines.