Pythagorean Triangle 3 : 4 : 5

W Bro Vic Brown PGStB

Many thanks to WBro Iain T PGStdB who forwarded the below for use in Devotion News. Iain is the L.E.O. at Baxter Lodge N^o 934, MAP presenter, and also member of the Kring Nieuw Holland Study Group.

Any triangle whose sides are in the ratio 3:4:5 is a right angle triangle. This has many occurrences in Freemasonry, but most obviously, in the Square.

After a Master installs his successor, a Senior Mason presents him with a Past Master’s jewel, which he is honoured to receive and wears with considerable pride. On the ribbon there is usually his Lodge logo and/or an engraved gilt bar, whilst below is a gilt square with a dangling bit. No one ever explains the significance of these to the proud owner. (Unless you hear the full charge given on its presentation, Ed).

THE SQUARE

The square can be one of two types; a symmetrical square similar to that on the Master’s Collar Jewel, or an asymmetrical square. It is the latter, which is of particular interest. Each leg is graduated, with one leg 3 units long, and the other is 4 units long, with the end scalloped. Each leg is 1 unit wide. Without the scallop, the distance between the insides of the ends would be 5 units.

This, of course, illustrates Pythagoras’s theorem; that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It is by no means certain that Pythagoras did come up with this proposition, but the Roman Architect Vitruvius, in his book, did credit Pythagoras for the idea. Vitruvius certainly made great use of the proposition in laying out and erecting many of his buildings.

However, it was not until 200 years later that Euclid finally proved Pythagoras’s Theorem with his 47th Proposition. It is the diagram of Euclid’s 47th Proposition which dangles from the PM jewel. A copy of Euclid’s Proposition is appended to this paper. 3 : 4 : 5 is the simplest form of the proposition, enclosing the largest relative area. The next combination of whole numbers is 5:12:13 and I have noticed that some Worshipful Masters on the Continent carry a large silver square on their Collars with that proportion – why, I have not been able to find out. The combinations of whole numbers beyond that all conform to the rule of the shortest side being an odd number and the other 2 having a difference of just 1.

It should be noted that whilst Vitruvius and Euclid gave the construction of the right angled triangle, they did not describe the remaining secrets of the 3 : 4 : 5 triangle known only to a Grand Master Mason. These will be illustrated later. So how does the 3 : 4 : 5 apply to us in Lodge?

On his Initiation, (in some Jurisdictions, not under UGLV) the Candidate is conducted to the East in readiness for his Obligation. He is instructed, “Left foot up the Lodge and right foot across the Lodge”. Note that originally this was not just heel to heel, but right heel directly behind the left heel, thus forming the 3 & 4 units of the triangle. “Take a short pace with your left foot, bringing the heels together, another a little longer and another longer still”. What is the significance of these increasing steps? I think some of you will already be ahead of me! Yes, it is the 3 : 4 : 5. By way of confirmation, in some Scottish Lodges, the Candidate is told to take a pace of about 9”, another of about 12” and another of about 15”, giving the ratio 3 : 4 : 5. In other Scottish Lodges he is told to take paces of 9”, 16” and 25” - 3 squared, 4 squared and 5 squared. So why do we start with the left foot? Formed in a square like this, the left foot is firmly in front and is the natural one to lead off.

THE 3 : 4 : 5 TRIANGLE

Let us now see where the 3 : 4 :5 triangle appears in our Craft Masonry. 2 triangles, end-to-end, give the proportion of the flaps on Craft, Mark and Royal Arch aprons.

Side-by-side gives the correct indentations around the Floor cloth – (demo) also the original indentations around the border of the Royal Arch apron, now slightly squashed up.

Only the 3 Grand Master Masons had possession of the 5, 4 and 3 unit rods, which were the secret for laying out buildings. Thus all three had to be present to form the right angle. ‘We three do meet and agree’.

We will now use the 5 point system to lay out a square building, such as a pyramid. (The rods are moved 45° and diagonal cords are laid out). The Grand Master Mason states the required equal distance from the centre to each corner, and these are marked. The four foundation lines are laid over the corner marks. These are called the ‘Land lines’. They are anchored beyond the corners, to act as sights, and are called ‘landmarks’ – the substantive origin of the word.

This square layout is geometrically straightforward, but now we will see one of the secrets of the 3 : 4 : 5 triangle. (The land-lines are removed and the rods are rotated to the next largest angle. The diagonals are again laid alongside the rods) The four land-lines are laid again equi-distant from the centre along the new diagonals – and we now have a floor plan with a ratio of length to breadth of exactly 2 to 1. See illustration below.

Finally we turn the rods to the smallest angle of the triangle, lay the diagonals along them, and then the land-lines on the 5 point system.

Lo and behold, we have the length to breadth ratio of exactly 3 to 1, generated entirely by the 3 : 4 : 5 triangle. So what was the size of King Solomon’s Temple? 60 cubits by 20! (3:1)

This ‘secret’ property of the 3 : 4 : 5 triangle was apparently unknown to Pythagoras and Euclid. Having laid out the plan, what happened before the 1st footing stone was laid? A human sacrifice was made. In those days it was considered an honour and they actually had volunteers! (Nowadays they are known as suicide bombers). 1st footing cornerstone is always laid in the N E. Why? In order to catch the first light of day at 6 am with the remainder following the sun, i.e. S E at 10 am, S W at 2pm and N W at 6 pm. Each corner had its individual human sacrifice. Later on, animals were sacrificed, and nowadays of course, time capsules are buried for posterity. The four corner stones were thus laid first and fully checked out. Only then were the remainder of the foundation stones laid in between. Incidentally, the emblems on a MM apron are usually but inaccurately described as ‘Levels’, despite not looking anything like a proper level as depicted on the SW’s pedestal. No, these emblems are the cross-section of the huge footing stones to take the massive weight of the finished building. Concrete had not been invented in those days. The corner footing stones were carved with this cross-section and it has been calculated that for a building the size of King Solomon’s Temple, each would have weighed 45 tons.

By now, you will have appreciated why in Lodge we follow the ancient custom of circumambulating the Lodge with the sun.

Subsequent courses were laid, one complete course at a time. Ell-shaped stones were used at the corners, alternating short ells and long ells. All of these stones were carved and aligned by hand with an accuracy which nowadays really beggars belief, especially realising the very heavy weights involved. Those who have taken the Nile trip from Luxor to Aswan, and seen all these temples, still standing after more than 3,500 years, without the use of any cement, will share my wonderment at the skill and knowledge of those ancient masons. They must have used certain techniques that have been lost in the passage of time.

Finally, the name of God used by our operative predecessors is ‘El Shadai’, as recognised by many Royal Arch Companions. You may be aware that all Hebrew characters have a dual function as both letters and numerals. The Hebrew characters for ‘El Shadai’ have the numerical value of 345. Is this just coincidence?

For some reason, of which I am unaware, our ancient brethren measured their pillars and columns by circumference, rather than by diameter. “The height of those pillars was 171⁄2 cubits, their circumference 12 and their diameter 4”. Similarly, the numerical value of the Hebrew characters for ‘Shadai’ by itself is 314 – the ratio of circumference to diameter or π. Is this just coincidence again?

Incidentally, King Solomon’s Temple had its main entrance in the East, with the Sanctum Sanctorum in the West, which was the norm for places of worship at that time. When the Christian Cathedrals came to be built however, the main entrance was located in the West. In like manner, the great majority of Christian Churches have their entrances in the SW.

Brethren, that completes my talk on 3 : 4 : 5 and I hope you will now look at the Past Master’s jewel in a different light, knowing that it represents much more than just ornamentation.