unit and gramme.

A very few denominations answer all the purposes of trade and commerce.

Herein lies the immense superiority of the decimal system over all others, because few subdivisions are necessary. As a system of currency the duodecimal and octonary systems have no advantage. The number of coins when halved and quartered are about the same under any system. We have 100, 50, 25, 10, 5 and I cent for decimal currency; and 80, 40, 20, 10, 5, I for octonary currency; and 60, 30, 15, 10, 5, I for duodecimal. Six different coins for each system, and one-half are common to all. We cannot do without the duodecimal and octonary systems of weights and measures, because we find it convenient to subdivide some things into twelve and others into eight parts, both in the workshop, warehouse, store and market. But neither the one system nor the other is adapted for countinghouse work; and, in a commercial community, counting-houses and banks form probably the most important branches of all commercial institutions. Nor are the duodecimal and octonary systems adapted for mental arithmetic and rapid calculations. when transacting business and estimating values in currency. The decimal system is essentially a currency system for banking and counting-house purposes. After we have made one binary, decimal, duodecimal, octonary and other divisions in the market, workshop, store and warehouse, in the practical operations of weighing and measuring, according to such tables of weights and measures as are in use, and we come to represent these weights and measures by values in coins and currency, we then want a simple, easy and rapid system to take up the values in weights and measures that needs but little reckoning and handling of figures. It is when all this has been done that the decimal notation steps in and asserts its preeminence over all others, for by this system the largest and most complicated fractional part can be expressed and computed as part of the integral factor as if no fraction was present.

The practical value and utility of any prime number as a metrical radix is oftentimes urged on purely mathematical grounds, and the numbers 12 and 8 are proposed because of the number of their subdivisions and general use in certain cases.

Thus 8 as a radix has given rise to the octonary system, and 12 to the duodecimal. But in a system of currency we don't look for a radix that can be bisected into the largest number of denominations; we look for a number as a radix that will multiply and divide quantities without any alteration of the figures, so as to save time and troublesome calculations and entries of figures of different denominations, and to prevent the chance of errors in computation. The decimal system is the only one that can do this. Whoever undertakes to devise a system of notation must remember that the very structure of all numbers implies a decimal notation; for the figures, as they stand in the order of rotation, are multiples of ten-as units, tens, hundreds, thousands, tens of thousands. A decimal notation is in the very structure of every number above ten.

Strathroy, Ontario, Canada.

S. BESWICK.

THE E. N. E. TRENCH AND OBLIQUITY OF THE ECLIPTIC.

The four great trenches on the east side of the Pyramid have attracted much attention. Found by Prof. C. Piazzi Smyth in the winter of 1864-5, they are now sufficiently investigated to warrant the belief that they held an important place in the construction of the wonderful monument with which they are connected. Some have maintained that they were only mortar troughs or reservoirs for water; others that they were connected with the azimuth and altitude of the Pyramid. I am not aware that an altogether satisfactory explanation of their use has yet been offered, and it may be that the following attempt at a solution of the question will not quite satisfy antiquarians, nevertheless the facts we here present may lead to the truth of the

matter.

may say that these trenches Close upon the east side of

Without being very exact, we are in length about 175 feet each. the Pyramid, and midway between the lines of its northern and

southern boundaries, a basalt pavement was constructed about 175 feet square. It was a magnificent work, which covered

more than one-third of an acre.

The blocks of basalt were all sawn and fitted together. Only one-quarter of it now remains in situ. Outside of this basalt square are the four great trenches, radiating, as it were, from its centre-one on the north side, beginning about 115 feet from the centre and running in a northerly direction; one on the south side, beginning about the same distance from the centre, and running in a southerly direction; and one on the east side, beginning about 90 feet from the centre and running E. N. E. The depth of these three trenches varies from 6 to 24 feet, and the breadth, near the surface of the rock in which they are cut, varies from 10 to 20 feet. The fourth trench, beginning at the very edge of the basalt square, runs N. N. E. with a gradual downward slope. It is about 3 feet wide and 20 inches deep. It has been thought that this shallow trench was built for water conduit to carry off the washings of the basalt pavement. But whether this be correct or not, such evidently was not the use of the three deep trenches we have described, for they were built with extreme care, even the natural rock excavation being cut to fit the stones with which they were lined, and have no water outlet as yet discovered.

Having thus given in a rough way the form and relative position of these trenches, we may present our theory of their use. The fact that all point inwardly towards the centre of the fine basalt square may be taken as evidence that they were constructed purposely to bear certain definite relations to one another and to the basalt paving, and thus to the Pyramid. To determine the character of these relations, it might serve us well if we could find some reason for the size and form of the square itself. We ask then, What was in the mind of the architect when he laid out this fine piece of paving?

The width from north to south through its centre is 2124.7 inches; the east side is located 2148. 3 inches east of the line of casing stones found by Mr. Petrie on the east side of the Pyramid. But these casing stones being 36 inches west of the meridian of the outer corner of the S. E. socket, the east side

of the basalt square is 2, 148. 3—36=2,112.3 inches east of that meridian. Therefore so much of this square as lies east of and contiguous to the meridian of the S. E. socket measures 2, 112.3 from west to east and 2, 124.7 from north to south. The polar axis of the earth being computed at 41,708,000 feet, one 20,000th part of it on a scale of one inch to a foot is 2,085.4 inches; and the equatorial axis in long. E. 31° being computed (Captain Clarke) at 41,852,000 feet, one 20,000th part of it, on the same scale, is 2,092.4 inches. Hence if an ellipse having a major axis equal to 2,092.4 inches and a minor axis equal to 2,085.4 inches were traced on the basalt square, as upon a drawingboard, tangent to the meridian of the S. E. corner socket, it would represent the meridional perimeter of the earth and leave a margin or border on the north, south and east sides equal to 19.6±1 inches, which is one-half the length of a second's pendulum in lat. 29° 58′ 51′′, and one-fourth the length of the coffer. From these evidences we may conclude that in laying out this basalt square the architect had in mind the equatorial and polar diameters of the earth.

Allowing this to have been the case, a legitimate inference is that the great trenches constructed north, south and east of the square, and pointing towards its centre, have an astronomical bearing upon the work and design of the Pyramid. Of the N. and S. trenches some uncertainty exists in Mr. Petrie's figures for the position of the inner end of the N. trench, owing to neglect to measure that end. According to his computations their axes are nearly parallel, but if they were designed to be meridional axes they would be distant from each other at the centre of the basalt square by 45.7 inches (one 100th of a half side of the geometrical base of the Pyramid), instead of 50 inches, as he gives it in his survey. However, the relation in which they stand to each other and to the basalt square, and the great probability that their axes were due N., appears to indicate that they were designed for transit observations, the bottom and sides being blackened and water introduced for reflection.

This hypothesis would greatly strengthen the theory that the E. N. E. trench was built for some astronomical use. Mr.

Petrie makes the azimuth of this trench 75° 58′ 23′′, and says "the axes at the ends were estimated by means of the plans here given, but on double this scale," which is, "and the rock is so roughly cut in most parts that nothing nearer than an inch need be considered." The trench has a narrow ledge at its east end, but along either side 140 feet westward the ledge is about 50 inches wide and 40 inches deep; the central or deeper part-that is, the trench proper-is 43 inches wide at the east end. The bottom or floor then is reached, by a somewhat abrupt descent, at 200 inches below the surface, thence it slopes downward 200 inches at an angle of about 20°, then runs level 300 or 400 inches, then slopes upward 300 or 400 inches at nearly the same angle as before, then gradually rises towards the surface at an angle of about 6°.

From the east end the sides of the deep part at first diverge rapidly and the trench widens as it approaches the basalt square, the widest place, however, being at the deepest part, or about 50 feet from the east end. Mr. Petrie found abundant evidence that this, as well as the north and south trenches, was lined with fine hard stone, "hardly less than 30 inches thick, considering the height was 20 feet" at the deepest part. Stones 10 or 15 inches thick would suffice for lining the shallow part of the trench towards the west end. This agrees well with the somewhat oval form of the rock cutting, as shown in the plan.

At 1,603 from the east end, where the bottom rises up to the level of the ledge, the whole width, including ledge, is 172. Supposing the ledge to be as wide there as at the east end, about 50, and the lining stones 10, a space of 52 inches would be left for the trench proper, 172-2 (50 + 10). This remarkable widening, in a direction opposite the sun rising,