Ratio ratio, that relation between magnitudes by which one is measured against another, is not a quantity, but a manner of comparison. Let two straight lines be given; let the first be divided into three equal parts, and the second into six. It is observed that the second contains twice the number of parts as the first. Thus, the ratio of the first to the second is as one to two. This relation holds whether the lines are long or short, so long as their parts are equal and their division is consistent. Ratio exists in the structure of the lines, not in their length alone. Let two other straight lines be given, unequal in length, yet such that the greater is divided into five equal segments, and the lesser into three. If the greater contains five parts, and the lesser three, then the ratio of the lesser to the greater is as three to five. This relation remains unchanged if both lines are doubled, or tripled, or multiplied by any number. For the parts, though increased, retain their proportion. The ratio is not altered by multiplication of the magnitudes, but only by the relative number of their parts. Let a straight line be cut at a point such that the whole is to the greater segment as the greater segment is to the lesser. This is the extreme and mean ratio. The whole line contains the greater segment, and the greater segment contains the lesser. When the parts are so arranged, the ratio of the whole to the greater equals the ratio of the greater to the lesser. This is not a common division. It is found only when the line is cut in a specific manner. Construct a line AB, and let it be cut at point C, so that AB is to AC as AC is to CB. Then, the square described upon AC is equal in area to the rectangle contained by AB and CB. This is demonstrated in the sixth book, proposition thirty. Let four magnitudes be given: A, B, C, D. If A is to B as C is to D, then the magnitudes are said to be in proportion. If A exceeds B in the same measure that C exceeds D, then the ratios are equal. If A is less than B, and C is less than D, and the excesses are in the same relation, then the proportion holds. This is not dependent on the size of the magnitudes, but on their mutual relations. One need not know the lengths, only how they compare. Let a triangle be drawn, and let a line be drawn parallel to one side, intersecting the other two sides. It is observed that the segments of those sides are in proportion. If the triangle has sides AB and AC, and a line DE is drawn parallel to BC, intersecting AB at D and AC at E, then AD is to DB as AE is to EC. This is not accidental. It follows from the definition of parallel lines and the nature of similar figures. The ratio persists, though the triangle be large or small, acute or obtuse. Let a circle be drawn, and let two chords intersect within it. The segments of one chord are to each other as the segments of the other. If chords AB and CD intersect at point E, then AE is to EB as CE is to ED. This relation holds for any such intersection, provided the chords lie within the same circle. The ratio is a property of the figure, not of its position or size. Let two squares be drawn, one with side three units, the other with side five. The area of the first is nine, of the second twenty-five. The ratio of their areas is as nine to twenty-five, which is the square of the ratio of their sides. This is true for all rectilinear figures. The ratio of areas is the duplicate of the ratio of corresponding sides. Similarly, for solids, the ratio of volumes is the triplicate of the ratio of corresponding edges. Ratio, then, is not a thing that can be touched, nor a number that can be counted. It is a relation that endures through multiplication, through division, through change of magnitude. It is not altered by the material from which the lines are drawn, nor by the surface upon which they are inscribed. It is a truth of space, independent of perception. Can the same ratio that divides a line into extreme and mean measure also be found in the arrangement of leaves upon a stem, or in the spirals of a shell? [role=marginalia, type=clarification, author="a.turing", status="adjunct", year="2026", length="49", targets="entry:ratio", scope="local"] Ratio is not merely numerical comparison—it is the invariant structure beneath scaling. One must not confuse the symbol with the relation: 3:5 persists whether lines are inches or parsecs. It is the computable essence of similarity, the precursor to measure, and the first whisper of algorithmic thought in geometry. [role=marginalia, type=clarification, author="a.darwin", status="adjunct", year="2026", length="41", targets="entry:ratio", scope="local"] Ratio is not merely numerical, but a fundamental expression of natural relation—observed in growth, form, and proportion across organic structures. Whether in leaf venation or bone segments, it is the constancy of relation, not absolute measure, that reveals nature’s underlying order. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:ratio", scope="local"]