What is the meaning of the Forty-seventh Problem of Euclid?

The Forty-seventh Problem of Euclid, also known as the Pythagorean proposition, states that in every right-angled triangle, the square erected on the hypotenuse is equal to the sum of the squares erected on the other two sides 2. Recognized in Masonic study as a cornerstone of mathematical science, this theorem provides the theoretical foundation for engineering, land surveying, navigation, and astronomy 212.

Within the emblem's moral and philosophical interpretation, it primarily instructs Masons to be general lovers of the arts and sciences 12. The emblem stands as a contemplation of absolute, omnipresent truth—a divine attribute that remains immutable across time and space 2. By enabling humanity to measure magnitude and reduce chaotic ignorance to cosmic law and order, the proposition serves as a symbol of the power, wisdom, and goodness of the Great Architect of the Universe, demonstrating that the book of Nature is to be read through a square 2.

Although traditional lecture text framed the problem as an invention of Pythagoras, commentary notes that the historical philosopher could not have received the Master Mason degree as it is known today 2. Mystical and historical explanations also connect the 3-4-5 triangle used in its demonstration to ancient Egyptian symbolism, where the base of three represented Osiris, the perpendicular of four represented Isis, and the hypotenuse of five represented Horus, illustrating how the product of two principles completes the figure 1.

In present-day practice, Quatuor Coronati Lodge records that the 47th Proposition of Euclid forms part of the jewel of a Past Master.

The jewel of a Past Master traditionally combines the square—the emblem of an Installed Master—with a representation of the 47th Proposition of Euclid. Under the United Grand Lodge of England rules, this jewel consists of the square and the diagram of the 47th proposition of the 1st Book of Euclid engraved on a silver plate pendant within it. Similarly, the Grand Lodge of Ireland explains that the Past Master's collar jewel pairs the square with the diagram illustrating the unique and perfect relationship between the sides of a right-angled triangle.

Freemasons adopted the proposition as a emblem instructing members to be devoted lovers of the arts and sciences 1. Recognized as one of the most important and celebrated theorems in mathematical science, the proposition serves as the foundation for practical geometry and trigonometry 1.

When suspended within the square on the Past Master's emblem, the diagram signifies that the Master who has completed his period of governance has grounded his leadership in mathematical truth, order, and dedicated study 1. The square represents morality and the authority of the Master's office, while Euclid's diagram represents the application of science and truth, together marking a ruler who has governed his lodge with skill and moral rectitude 1.

Pythagoras is traditionally linked to the Forty-seventh Problem of Euclid as its discoverer, though Masonic commentary distinguishes between historical reality and monitorial legend 2. Monitorial lectures derived from Thomas Smith Webb describe Pythagoras as an ancient friend and brother who discovered the theorem during his travels, exclaimed "Eureka," and sacrificed a hecatomb in exultation 2. Carl H. Claudy notes that while Pythagoras was a great philosopher and mathematician, his raising as a Master Mason is a historical impossibility, and the sacrifice of a hundred cattle contradicts the vegetarian tenets of the Pythagoreans 2. Albert C. Mackey adds that Pythagoras likely acquired the geometric principle from Egyptian priests, who used a three-four-five right triangle to symbolize Osiris, Isis, and Horus 1. Euclid of Alexandria later compiled the existing geometrical knowledge of his era, recording the proposition at the close of his first book two centuries after Pythagoras 2.

Adopted into the Master Mason degree, the emblem of the Forty-seventh Problem instructs members to be general lovers of the arts and sciences 12. Operatively, the theorem serves as the cornerstone of mathematical science, providing the calculation necessary for architecture, land surveying, engineering, and astronomy 2. Symbolically, the problem represents immutable and omnipresent truth, reflecting the order of creation and serving as an emblem of the power, wisdom, and goodness of the Supreme Architect of the Universe 2.