The 47th Problem of Euclid

A Symbol of Geometry; of exact science. Passed over with but a few words of ritual, it is Masonically most interesting. It appears on the frontispiece of Anderson's Constitutions, published in 1723; Street states it is the earliest example of a printed symbol of Freemasonry. It was apparently known to ancient mathematicians long before Pythagoras (Masonically credited as its discoverer) or Euclid, who made the properties of a right angled triangle his forty-seventh problem. It is the root of all mathematics used to determine an unknown from two knowns. Given the distance of a mountain and the angle of sight to its top, mathematics may determine its height. Tunnels are driven through mountains from both sides to meet exactly by means of meansurements made by the forty seventh problem. Navigation of the seas depends upon it. In non-Euclidian language, a right angled triangle of 3 feet base, and 4 feet height, has a line 5 feet long joining the free ends of two legs. The square of 3 is 9; the square of 4 is 16; the sum of 9 and 16 is 25; the square root of 25 is 5.

All right angled triangles, regardless of the length of base and upright, follow this law; that the line joining the free ends (the hypotenuse) is the square root of the sum of the squares of the two sides. Therefore, if any two of the three are known, the third may be calculated.

According to the ritual "it teaches Masons to be general lovers of the arts and sciences". This short description encompasses the study of Geometry.

Masonic Service Association

In 2003, R.'.W.'. Lee Miller, a noted author and freemason, created a proof the the 47th Problem incorporating the concept of the Divine Proportion, or "Golden Section".

As you can see in the diagram above, the bottom square is bisected by the line at the hypotenuse- creating an exact golden section. This section is "1.618" on one length, and "1" on the other. The neat thing about this is that the 1:1.618 ratio is known as universally pleasing, a harmonious proportion, golden or Divine in nature. In fact, it appears in nature regularly, showing up in the webbed structure of leaves, heights of tree structures, lengths and facial proportions in animal forms, sea shells (The Nautilus), classical art composition (Rembrandt, Titian and other old masters), musical scale structure and notation, and even the architecture of the Pyramids. The Divine Proportion also shows a perceived harmony of our own human anatomy:

In the famous diagram above, drawn by our friend Leonardo DaVinci, the human form is broken into several different examples of the Divine Proportion, and also fits perfectly within a square and circle. (Interestingly enough, this diagram was used on the original Voyager I spacecraft to exemplify human form and proportion. The Voyager I mission was a probe sent from Earth in the 1970's to journey towards and beyond the ends of our Solar System, to send information back to us as long as it was within range, and also act as an early emissary into the Galaxy, a representative of human life, just in case. See it in the movie Star Trek I! )Leonardo's exemplary diagram clearly illustrates our human form and the Divine Proportion working its way in the ratio of our body to legs, arms to torso, and even in the placement of our facial features.

So... these two items, the "Divine Proportion" and the "47th Problem" each contain a mathematical pin-point of "divine light", a physical constant or limitation that The Great Architect, through nature, uses for structure. A clue!

R.W. Miller is the author of "Freemasonry and a View of the Perennial World Philosophy," and founder and Director of the Onondaga and Oswego Masonic Districts Historical Societies. They have a website at . And there are further resources available at If you would like to contact Bro. Lee Miller, his email is

An excellent book about the divine proportion is Gyorgy Doczy's "The Power of Limits - Proportional Harmonies in Nature, Art, and Architecture."