The Treatise of Measures (Mishnat ha-middot)

Chapter 1

Areas can be classified in four ways, and they are as follows: the quadrilateral, the trilateral, the circle, and the arc. This is the general rule: the second is half of the first, while the fourth is half of the third. The others are intertwined with each other, like a girdle in a band.

The quadrilateral has three facets: the side, the rope, and the roof. What is the side [tsela‘]? It is that which holds the walls of the roof, as it is stated: the altar shall be square . . . and the bars shall be upon the two sides [tsela‘ot] of the altar (Exodus 27:1,7). And the rope is that which makes a break from one corner to the other corner, from end to end (Exodus 26:28), in the longest manner along the length of the roof. The roof is itself the area.

The trilateral has three facets: the two sides and the base, the post, and the roof. What are the two sides? These are the two extensions, right and left, as it is stated: For you shall spread abroad to the right and to the left (Isaiah 54:3). The base is that upon which the two sides are affixed, as it is stated: [the pillars] upon which the house rests (Judges 16:26). The post is the shared line that descends from between the two sides to the base. It is in the corner [as in]: for the corners of the Tabernacle (Exodus 26:23). The roof is itself the area.

The circle has three facets: the circumference, the rope, and the roof. What is the circumference? It is the line that compasses the circle, as it is stated: And a line of thirty cubits compassed it round about (1 Kings 7:23). The rope is that which extends from edge to edge, as it is stated: from brim to brim (2 Chronicles 4:2). The roof is itself the area.

The arc has four facets: the bow, the cord, the arrow, and the roof. What is the bow? Part of the circle, as it is stated: like the appearance of the bow that is in the cloud (Ezekiel 1:28). The cord is that which holds the opening of the bow, as it is stated: the bent bow (Isaiah 21:15). The arrow is that which extends from the middle of the bow to the middle of the string, as it is stated: [The wicked bend the bow,] they have made ready their arrow upon the string (Psalms 11:2). The roof is itself the area.

How does one calculate the area? You multiply one upon the other. This is the area, and it is one cubit by one cubit. If the roof is equal in sides and corners, you count them from each side. If the standing board [square] is comprised of two on each side, and the corners are equal, then the area holds four times the area of the unit, which is one cubit by one cubit. If it is comprised of three on each side, then it is nine times the area of the unit, and similarly with regard to four by four and five by five. From here on, you can go and compute upward, according to this method.

You divide those that are less than a single unit in the following manner: one cubit into two ropes, breaking [intersecting] each other in the middle, from the right side to the left side, and likewise from the top to the bottom. The result is that the roof [area] is divided into four compartments, and you will find that each is half a cubit by half a cubit, while the area itself is a quarter of a cubit, which is a quarter on each side. The same applies to a third by a third, and a fifth within a fifth, with regard to both equal and unequal sides. From here on you can go and compute fractions downward, according to this method.

They have already said: half by a half is a quarter, and similarly a third by a third is a ninth. In these cases, and in other such instances, only with equal and unequal sides, you count them from the primary, which means that you count as follows: ten by ten make a total of one hundred; half of ten is five; five times five is twenty-five, which is a quarter of one hundred. The status of the ten is in the one, the status of the one hundred is in the ten, and the one thousand is in the one hundred. From here on you can go to compute fractions in accordance with the measure of the integers, only that in the case of the integers it increases whereas in the case of the fractions it decreases.

This is the general rule: half by half is a half of the half; and a third by a third is a third of the third. Similarly, a half by a third is a half of the third, and likewise a quarter by a third is a quarter of the third, with regard to these and other such instances, with equal and unequal sides.

Chapter 2

One who wishes to measure quadrilateral fields, with equal or unequal sides, should calculate the length by the width, and the area is the result of them both.

In the case of a trilateral, whether it has equal or unequal sides, one should calculate the post by half of the base, and the area is the result of them both. There are many paths to reach this.

How is it for a circle? One should calculate [multiply] the rope [radius] with itself [i.e., square it], and take away from it one seventh and half of one seventh. The remainder is the area. For example, if a rope extends to seven, its sum is forty-nine; one seventh and half of one seventh is ten and a half. Thus, the area is thirty-eight and a half.