Composition of ratios

Composition of ratios is the rule by which several successive comparisons between magnitudes of the same kind are treated as one compound comparison. Its decisive importance is that it converts a chain of local relations into a single relation between the first and last terms, giving proportional reasoning a precise structure that can be demonstrated geometrically and clarified algebraically.

Definition and Structure

The treatment begins with a formal definition of a compound ratio. When several magnitudes of the same kind are arranged in sequence, the first is said to have to the last the ratio compounded of the ratio of the first to the second, the ratio of the second to the third, the ratio of the third to the fourth, and so on until the final magnitude. For four magnitudes A, B, C, and D, the ratio of A to D is therefore understood through the successive ratios A to B, B to C, and C to D. The compound relation is not an unrelated new comparison; it is the organized result of the intermediate comparisons that connect its extremes. The same definition also permits one ratio to be replaced by another equal ratio. If several corresponding ratios are equal respectively, the ratio formed from one set may be described as compounded of ratios that are the same as the ratios in another set. This makes composition a relational operation: what matters is not the particular names of the magnitudes but the equality of the component ratios and the order in which they are combined.

Algebraic Interpretation

The work makes the definition more intelligible through an arithmetical or algebraical illustration. A ratio compounded of several ratios is presented as having, for its antecedent, the continued product of all the antecedents of the component ratios, and, for its consequent, the continued product of all their consequents. Thus the operation can be represented by multiplying corresponding first and second terms before comparing the resulting products. The illustration gives the component ratios 2:3, 4:5, 6:11, and 2:5; their compound ratio is formed as 2 multiplied by 4, 6, and 2 compared with 3 multiplied by 5, 11, and 5, yielding 96:825, which reduces to 32:275. A parallel formulation is given for magnitudes A, B, C, D, E, and F. The ratio A:F is compounded of A:B, B:C, C:D, D:E, and E:F. In this expression the intermediate terms appear in both the products associated with the successive ratios, so the chain can be understood as a systematic cancellation of adjoining terms, leaving the relation between the initial magnitude and the final one. The algebra does not replace the definition; it exposes the mechanism by which the successive comparisons cohere.

Relation to Geometrical Demonstration

Composition of ratios is situated within the demonstrative architecture of geometry. Definitions state what the relevant objects and relations mean; postulates grant permitted constructions, such as drawing a straight line from one point to another or describing a circle with a chosen centre and radius; axioms state general truths accepted as self-evident; and demonstration proves propositions by reasoning from these foundations. A theorem asserts the truth of a principle, while a problem proposes an operation to be carried out with a Straightedge and Compass. Composition of ratios belongs to this same discipline of moving from established relations to a justified conclusion rather than relying on appearance alone. The distinction between visual material and mathematical reasoning is especially important in Oliver Byrne's edition. Geometrical diagrams are described as the materials of geometrical science, whose purpose is to show the relative quantities of their parts through a reasoning process called demonstration. The diagram makes relations available to sight, but the proof establishes what follows from them. In that setting, a compound ratio is not merely a multiplication rule borrowed from arithmetic: it is a way of preserving the logical connection among geometrically related magnitudes.

Coloured Representation and Cognitive Use

The book's coloured system gives proportional reasoning a visual grammar. Ordinary letters attached to points, lines, or other parts of a diagram are arbitrary names used in a demonstration; differently coloured parts are intended to name themselves, because corresponding forms and colours represent them throughout the reasoning. The aim is speed and precision: the reader can identify the relevant part by its visible colour instead of tracing a sequence of letters and reconstructing the intended magnitude mentally. This visual economy is also applied to equality. When parts that are given as equal have the same colours, the arrangement provides an ocular demonstration of the parts to be proved equal and helps the learner retain the data during the reasoning. In oral teaching, colour can be presented to the eye while the explanation reaches the ear, so a red angle, blue line, or other coloured component is immediately located by the class. The system is offered as an auxiliary as well as a substitute for ordinary lettering, useful for introduction, rapid reminiscence, and more permanent retention. The author nevertheless places a strict limit on what colour means. A mathematical line is length without breadth and cannot literally possess colour. The junction of two colours on the same plane gives a familiar representation of such a line, but when a text speaks of a black line or a red line, the intended object is the junction and the geometrical relation, not the colour as a physical property of the magnitude. Colour functions as a name and a guide to attention; it does not alter the line, angle, or magnitude being reasoned about.

Place in the Theory of Knowledge

Composition of ratios illustrates the book's broader attempt to make mathematical knowledge both formally secure and perceptually accessible. The text distinguishes self-evident axioms from propositions obtained by reasoning and insists that a demonstration prove that the indicated process really reaches its required end. Its coloured diagrams therefore do not make proof unnecessary. They are designed to prevent the learner from losing the data, confusing one part with another, or reducing the demonstration to a memorized verbal sequence. The method also reflects the work's educational ambition. It claims that the coloured presentation can reduce the time required to acquire the Elements of Euclid to less than one third of the usual period and make retention more permanent, based on experiments by the inventor and others who adopted the plan. Whether one accepts the full scale of that claim or not, the underlying principle is clear: the composition of ratios should remain visible as a chain of relations, while formal reasoning explains why the chain yields its final comparison. The concept thus joins exact proportion, demonstrative logic, and a theory of learning in which the eye helps preserve the structure that the proof establishes.