Commentaria in Numeros
Where it appears that each side of the suburbs is twice as long as the line drawn outward from the city, which is of a thousand cubits (verse 4), and so to the several sides two thousand cubits are left.” The first objection of our Cornelius is gratuitous; to the second it is answered, both that the cities of the Levites were quite small, and that Moses in defining numbers does not always use mathematical accuracy, as they say, nay often uses round numbers for imperfect ones.
Fourthly, [we must look] not to the Hebrew text, but to our Interpreter:
For these suburbs contained a thousand paces, and a thousand paces make two thousand cubits. For although a pace is greater than a cubit, and contains five feet, when we measure a fixed pace, namely both feet of the measurer and the three feet between his two feet intercepted in the pace, yet if you measure and take a moving pace (such as this was) and continuously advancing, you will find it contain only four feet. For the first foot is counted only in the first pace, so that this first pace has five feet; in the following paces only the remaining four feet are counted, continuously succeeding one another, as is plain to one who tries; but four feet make two cubits, and a little more. Therefore these thousand paces are the same as the two thousand cubits of which in the following verse. Vilalpandus adds, part II On the City and Temple, book II, chapter i, that the common cubit is a foot and a half, but the sacred cubit is two feet and a half, so that two sacred cubits make a pace, that is five feet.
You will say: In the Hebrew, both in this verse and in the following, the same word amma stands, which signifies a cubit: how then does our Interpreter render it here a pace, but in the next verse a cubit?
It might be answered that amma signifies not only a cubit but also a pace: for this not only our Interpreter teaches, but Eugubinus also; and that the Interpreter renders the same word amma first a pace, then presently a cubit, is no wonder, because he saw that the circumstances of the place required it; otherwise there would be a manifest contradiction here. When, therefore, it is said in the Hebrew simply a thousand amma, he understood amma absolute and greater, that is a pace to be signified; but when presently it is said restrictively, with the letter beth, two thousand baamma, that is in amma, he understood amma contracted and less, namely a cubit. Perhaps also in the former place, namely this verse, instead of amma he read paam, that is a step, a pace, so that the letter pe in the codex being worn or eaten away, there remained am, and thence the copyists made amma: for they knew that not am, but amma, signified a certain measure. But because the Hebrews constantly have amma, with aleph, not with ain, which in Scripture always signifies a cubit, never a pace, and so here the Chaldee, the Septuagint and all other interpreters render it, I should rather say that our Interpreter here takes pace, not the greater or geometrical, which comprises five feet, but the lesser or vulgar, so that it is the same as a cubit. For in walking the vulgar pace, such as that of those who walk slowly and modestly, is equal to a cubit, if you take from it the first foot, on which you stand, and measure only the space between the two feet (which is commonly half a foot), and the foot itself which you extend and advance in this pace. For so this lesser pace contains about a foot and a half; and a cubit contains as much.
You will say: How then in this verse are the suburbs numbered to a thousand paces or cubits, but in the following verse to two thousand cubits? I suppose that both the cities and these suburbs were of circular or oval form.
Now I say: This antilogy can be solved and reconciled in two ways: first, if with Cajetan we say that these suburbs were of a thousand cubits, but that their circuit or periphery was about eight thousand cubits; so that this periphery from any corner of the world, that is any quarter, had two thousand cubits.
For which note: In a circle the circumference has a triple proportion to the diameter, and a little more, so that if the diameter be of two thousand cubits, the circumference is of six thousand cubits. But here the semi-diameter of the suburbs was a thousand cubits: add to this the diameter of the city itself, which was about six hundred cubits, so that the diameter both of the city and of the suburbs was 2600 cubits; from which it necessarily follows that the circumference (which is triple to the diameter) both of the city and of these suburbs was about eight thousand cubits; for three times 2600 make eight thousand, less two hundred, which must be added here, because the circumference is a little more than triple to the diameter: now divide eight thousand by the four quarters of the world, you will find the fourth part of the circumference of the suburbs, which faced each quarter, to be about two thousand cubits. And this is what is said in the following verse, that toward each quarter there were two thousand cubits.
“Moses,” says Cajetan, “established the circumference of the suburbs at eight thousand cubits, whether the city be great or small, that there might be one common law concerning the space of the suburbs. For he judged it equitable that the diameter of any city inside be held to be six hundred cubits; for although one city was greater than another, yet, balancing the greater with the less, and conversely, he established equal suburbs; for the diameter being 2600 cubits, equally according to geometrical proportions the circumference is determined to be eight thousand cubits, according to the triple proportion and a little more.” But I answer, first, that the angle (as it is in the Hebrew) of each quarter in the following verse is commanded to be drawn out to two thousand cubits, and because angle properly signifies not the periphery of a circle, but an angle or triangle, and because God commands precisely a thousand paces, that is cubits, to be reckoned outside each city; but some cities were great, others small, and consequently some had less, some more than eight thousand cubits in the periphery of the suburbs: Hence secondly, you may more aptly take these two thousand cubits not in the periphery, but in the lines from the centre, namely drawn from the walls of the city to the periphery of the suburbs; for these lines, since there are two toward each quarter of the world, that they may between them take in and comprehend it whole, make one angle at the centre where they meet, and two angles at the periphery where they end, and so make a triangle. Whence you may render the Hebrew literally thus: you shall measure at the angle of the East two thousand cubits; as many at the angle of the South, of the West and of the North.
For God seems here to describe the circle of the city with its suburbs, so that the centre is the city, and the periphery ends at a thousand cubits round about outside the walls of the city. This circle He divides, toward the four quarters, into four as it were triangles, each of which is isosceles, that is, has the two sides drawn from the centre to the periphery equal. God therefore here commands that the suburbs be extended in every direction, toward the four quarters of the world, to a thousand cubits, and that the eastern quarter be described by two lines drawn from the city to the periphery of the suburbs, which two lines comprehend the eastern side, after the manner of a triangle. In like manner, He commands a second triangle to be described by two lines at the western quarter, and in like manner a third triangle at the south, and a fourth at the north; so that each quarter is described by a triangle, two lines being drawn from the city to the periphery aforesaid, each of which contains a thousand cubits (for the suburbs are extended to a thousand cubits), so that the two lines drawn to the periphery of any quarter, which are the two equal sides of this triangle, together contain two thousand cubits.
Thus, then, we easily reconcile this antilogy; for we say that the suburbs were extended only to a thousand cubits round about, but that every quarter was described as it were by a triangle, so that it had two sides, each of which was of a thousand cubits, and consequently both sides taken together were of two thousand cubits. For it is likely that eight limits or stakes of these suburbs were set at their very borders, so that from the city to them eight lines in all could be drawn, which make four triangles as it were; and this to the end that by these limits it might be at once known all round how far the lands of the Levites extended, and so the frequent quarrels which are wont to arise about boundaries with neighbouring tribes might be avoided. In favour of this sense are the Hebrew words, which have, thou shalt number two thousand in the cubit; which phrase intimates that these two thousand cubits are to be numbered, not directly in length, but laterally, namely along the two sides of the triangle, so that each side has a thousand cubits, as may be seen in the following figure: