Unit I: Introduction, Definitions, Postulates, Axioms & Symbols¶
Oliver Byrne — Pickering Edition (pp. vii–xxix, Leaves 11–33)
Introduction: The Ocular Manifesto (pp. vii–xvi)¶
The arts and sciences have become so extensive, that to facilitate their acquirement is of as much importance as to extend their boundaries. Illustration, if it does not shorten the time of study, will at least make it more agreeable. This Work has a greater aim than mere illustration; we do not introduce colours for the purpose of entertainment, or to amuse by certain combinations of tint and form, but to assist the mind in its researches after truth, to increase the facilities of instruction, and to diffuse permanent knowledge.
In accordance with Horace's maxim (Ars Poetica 180–182) and the intuitive pedagogy of Pestalozzi, Oliver Byrne states on page ix the foundational principle of pointer-free visual notation:
“In the demonstrations of Euclid, the letters annexed to points, lines, or other parts of a diagram are in fact but arbitrary names, and represent them in the demonstration; instead of these, the parts being differently coloured, are made to name themselves, for their forms in corresponding colours represent them in the demonstration.”
On pages x–xi, Byrne contrasts six elementary relations of a right-angled triangle (legs , , hypotenuse ) expressed first in thirty lines of alphabetic prose (AB, BC, AC, angle BAC) and then in six immediate colored equations:
+ + =
⊏ and ⊏
+ ⊏
2 = 2 + 2
Book I. Definitions (I–XXXV, Simson Numbering, pp. xvii–xxii)¶
I. A point is that which has no parts.
II. A line () is length without breadth.
III. The extremities of a line are points.
IV. A straight or right line () is that which lies evenly between its extreme points.
V–VII. A superficies (surface) has length and breadth only; its extremities are lines; a plane superficies is that in which any two points being taken, the straight line between them lies wholly in that plane.
VIII–IX. A plane angle () is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction; a plane rectilineal angle is contained by two straight lines.
X. When a straight line standing on another straight line makes the adjacent angles equal to one another ( = ), each of the angles is called a right angle (); and the straight line which stands on the other is called a perpendicular (⊥) to it.
XI–XII. An obtuse angle is greater (⊏) than a right angle; an acute angle is less (⊐) than a right angle.
XV–XVIII. A circle () is a plane figure contained by one line, called the circumference, such that all straight lines drawn from a certain point within the figure (the centre) to the circumference are equal to one another; a diameter passes through the centre and bisects the circle into two semicircles.
XIX–XXIII. Rectilineal figures: trilateral (), quadrilateral (), and multilateral polygons.
XXIV–XXVI. Among trilateral figures: an equilateral triangle has three equal sides; an isosceles triangle has two equal sides; a scalene triangle has three unequal sides; further divided into right-angled, obtuse-angled, and acute-angled triangles.
XXVII–XXXIV. Among four-sided figures (Simson's expanded classification): XXVII. a square () has all its sides equal and all its angles right angles; XXVIII. an oblong () is right-angled but not equilateral; XXIX. a rhombus is equilateral but not right-angled; XXX. a rhomboid has opposite sides equal without being equilateral or right-angled; XXXI–XXXIV. trapezia and parallelograms.
XXXV. Parallel straight lines (∥) are such as are in the same plane, and which being produced ever so far both ways do not meet.
Postulates (I–III) & Axioms (I–XII, pp. xxiii–xxiv)¶
Postulates
Postulate I ([I.post.1] /
G.24): Let it be granted that a straight line () may be drawn from any one point to any other point.Postulate II ([I.post.2] /
G.25): That a terminated straight line may be produced to any length in a straight line ().Postulate III ([I.post.3] /
G.26): And that a circle () may be described from any centre, at any distance from that centre.
Axioms (Simson Numbering)
Axiom I ([C.N.1]): Things which are equal to the same thing are equal to one another.
Axiom II ([C.N.2]): If equals be added to equals, the wholes are equal.
Axiom III ([C.N.3]): If equals be taken from equals, the remainders are equal.
Axioms IV–V: If equals be added to or taken from unequals, the wholes or remainders are unequal (⊏ / ⊐).
Axioms VI–VII: Things which are double of the same, or halves of the same, are equal to one another.
Axiom VIII ([C.N.4]): Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
Axiom IX ([C.N.5]): The whole is greater (⊏) than its part.
Axiom X: Two straight lines cannot enclose a space.
Axiom XI ([I.post.4] /
G.27): All right angles () are equal to one another.Axiom XII ([I.post.5] /
G.28): If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less (⊐) than two right angles (), these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.
Symbols and Abbreviations (pp. xxvii–xxix)¶
Glyph |
RST Role |
Meaning in Byrne (1847) |
|---|---|---|
|
|
Deductive inference (therefore) and premise (because). |
|
Standard operators |
Equality, geometric composition (sum), and remainder (difference). |
⊏ / ⊐ |
|
Greater than ( |
∥ / ⊥ |
|
Parallel straight lines and perpendicularity. |
|
One right angle (quadrant arc between perpendicular arms). |
|
|
Two right angles (bisected semicircle standing on a straight line). |
|
2 |
|
Square described upon the colored straight line. |
∷ |
|
Equality of ratios ( |