[44.] Sections 44 and 45 are from MS. Digby 72. This long explanation of the method of finding a planet's place depends upon the tables which were constructed for that purpose from observation. The general idea is this. The figures shewing a planet's position for the last day of December, 1397, give what is called the root, and afford us, in fact, a starting-point from which to measure. An 'argument' is the angle upon which the tabulated quantity depends; for example, a very important 'argument' is the planet's longitude, upon which its declination may be made to depend, so as to admit of tabulation. The planet's longitude for the given above-mentioned date being taken as the root, the planet's longitude at a second date can be found from the tables. If this second date be less than 20 years afterwards, the increase of motion is set down separately for each year, viz. so much in 1 year, so much in 2 years, and so on. These separate years are called anni expansi. But when the increase during a large round number of years (such as 20, 40, or 60 years at once) is allowed for, such years are called anni collecti. For example, a period of 27 years includes 20 years taken together, and 7 separate or expanse years. The mean motion during smaller periods of time, such as months, days, and hours, is added in afterwards.
[45.] Here the author enters a little more into particulars. If the mean motion be required for the year 1400, 3 years later than the starting-point, look for 3 in the table of expanse years, and add the result to the number already corresponding to the 'root,' which is calculated for the last day of December, 1397. Allow for months and days afterwards. For a date earlier than 1397 the process is just reversed, involving subtraction instead of addition.
[46.] This article is probably not Chaucer's. It is found in MS. Bodley 619, and in MS. Addit. 29250. The text is from the former of these, collated with the latter. What it asserts comes to this. Suppose it be noted, that at a given place, there is a full flood when the moon is in a certain quarter; say, e.g. when the moon is due east. And suppose that, at the time of observation, the moon's actual longitude is such that it is in the first point of Cancer. Make the label point due east; then bring the first point of Cancer to the east by turning the Rete a quarter of the way round. Let the sun at the time be in the first point of Leo, and bring the label over this point by the motion of the label only, keeping the Rete fixed. The label then points nearly to the 32nd degree near the letter Q, or about S.E. by E.; shewing that the sun is S.E. by E. (and the moon consequently due E.) at about 4 A.M. In fact, the article merely asserts that the moon's place in the sky is known from the sun's place, if the difference of their longitudes be known. At the time of conjunction, the moon and sun are together, and the difference of their longitudes is zero, which much simplifies the problem. If there is a flood tide when the moon is in the E., there is another when it comes to the W., so that there is high water twice a day. It may be doubted whether this proposition is of much practical utility.
[41a]: This comes to precisely the same as Art. 41, but is expressed with a slight difference. See [fig. 16], where, if bc = 8, then BC = 12⁄8 EB.
[41b]: Merely another repetition of Art. 41. It is hard to see why it should be thus repeated in almost the same words. If bc = 8 in [fig. 16], then EB = 8⁄12 BC = 2⁄3 BC. The only difference is that it inverts the equation in the last article.]
[42a] This is only a particular case of Art. 42. If we can get bc = 3, and b′c′ = 4, the equations become EB = 4BC, E′B = 3BC; whence EE′ = BC, a very convenient result. See [fig. 17].]
[43a]: The reading versam (as in the MS.) is absurd. We must also read 'nat come,' as, if the base were approachable, no such trouble need be taken; see Art. 41. In fact, the present article is a mere repetition of Art. 43, with different numbers, and with a slight difference in the method of expressing the result. In [fig. 18], if b′c′ = 3, bc = 4, we have E′B = 3⁄12 BC, EB = 4⁄12 BC; or, subtracting, EE′ = (4-3)/12 BC; or BC = 12 EE′. Then add the height of E, viz. Ea, which = AB.
[42b.]: Here, 'by the craft of Umbra Recta' signifies, by a method similar to that in the last article, for which purpose the numbers must be adapted for computation by the umbra recta. Moreover, it is clear, from [fig. 17], that the numbers 4 and 3 (in lines 2 and 4) must be transposed. If the side parallel to bE be called nm, and mn, Ec be produced to meet in o, then mo : mE :: bE : bc; or mo : 12 :: 12 : bc; or mo = 144, divided by bc (= 3) = 48. Similarly, m′o′ = 144, divided by b′c′ (= 4) = 36. And, as in the last article, the difference of these is to 12, as the space EE′ is to the altitude. This is nothing but Art. 42 in a rather clumsier shape.
Hence it appears that there are here but 3 independent propositions, viz. those in articles 41, 42, and 43, corresponding to figs. 16, 17, and 18 respectively. Arts. 41a and 41b are mere repetitions of 41; 42a and 42b, of 42; and 43a, of 43.