Duplicate ratio

Duplicate ratio is a state of proportional relation defined through three magnitudes arranged in continued proportion. Its decisive feature is that the ratio of the first magnitude to the third is the square of the ratio of the first to the second, so a single comparison is transformed into a compounded, second-degree relation.

Definition and Structure

When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second. The definition depends on the order of the magnitudes: the first-to-second comparison supplies the original ratio, while the first-to-third comparison expresses its duplication. Duplicate ratio therefore does not mean merely that two ratios happen to be equal; it records a specific proportional sequence and the relation produced by extending it.

The underlying arrangement is a chain of continued proportionals. If the magnitudes are represented as A, B, and C, then A bears to B the same ratio that B bears to C. In that arrangement, A is said to have to C the duplicate ratio of A to B. The third magnitude is thus not an unrelated reference point but the final term through which the original ratio becomes squared.

Algebraic Meaning

The text expresses the relation by stating that the ratio of A to C is the square of the ratio of A to B. In symbolic form, the duplicate ratio is therefore a squared ratio rather than a simple repetition of the initial comparison. The definition connects geometric proportionality with multiplication: the continued proportion supplies the repeated factor, and the comparison between the outer terms displays the resulting square.

A numerical pattern makes the structure more visible. If A, B, and C are continued proportionals, the relation may be written as A : B :: B : C; the ratio A to C then represents the square of A to B. The same point is restated through quantities represented by a and r: the proportional continuation produces a squared expression, and the ratio of the outer terms equals the square of the ratio of the adjacent terms. The repeated algebraic formulation reinforces that duplicate ratio is an operation on a ratio, not simply a name for a visual resemblance.

Relation to Geometrical Demonstration

Within the broader method of the book, this definition belongs to a system in which mathematical relations are made more immediate through diagrams, colours, and carefully assigned symbolic roles. The book distinguishes the geometrical diagram itself from the representative signs used to discuss it: the parts of a figure are the materials of geometrical science, while letters or colours identify those parts during demonstration. Duplicate ratio follows the same principle of controlled reference, though its essential relation is proportional and numerical rather than merely visual.

The coloured system is presented as a way to make corresponding parts identify themselves more rapidly than sequences of arbitrary letters. In an oral demonstration, a coloured angle or line can be indicated at once to an entire class, while a lettered designation requires the students to trace the letters and reconstruct the intended part. This educational claim illuminates the presentation of proportional concepts as well: the aim is to keep the relevant terms and their relations continuously before the learner rather than leave them as disconnected verbal formulas.

Place in the Mathematical System

Duplicate ratio depends on the same formal discipline that governs the book's definitions, postulates, and axioms. A point is defined as that which has no parts; the postulates grant, among other things, that a straight line may be drawn from any one point to any other and that a circle may be described with any centre at any distance from that centre; the axioms state general principles about magnitudes, including the relation of magnitudes equal to the same magnitude. These foundations establish the habit of separating primitive assumptions, definitions, and demonstrated consequences.

The concept also stands beside the book's account of superposition, by which one magnitude may be conceived as placed upon another so that corresponding parts exactly coincide. Such procedures make equality and comparison intelligible through controlled correspondence. Duplicate ratio extends that comparative logic from direct coincidence or equality to a proportional relation in which the outer terms encode the square of an inner ratio.

Educational Significance

The book claims that coloured demonstrations can be learned in less than one third of the time ordinarily required and retained more permanently, attributing these results to experiments conducted by the inventor and others who adopted the method. It also argues that simultaneous appeal to sight and hearing is especially advantageous in classroom teaching, because the spoken explanation and the visible diagram reinforce one another at the same moment.

This approach opposes memorizing demonstrations without understanding. When equal parts receive the same colours, the arrangement is described as an ocular demonstration that keeps the given data before the learner throughout the reasoning. Duplicate ratio, with its repeated proportional structure and squared outcome, exemplifies the kind of relation that benefits from such continuity: the learner must preserve the order of the three magnitudes while following how the initial ratio is carried through the proportion to the outer terms. Colour does not belong mathematically to a line, angle, or magnitude; it functions only as a name for the relevant part. The mathematical content remains the proportional relation itself.