multiply XCVIII by MDLVII, without using Arabic or common numerals. Professor McArthur, in his article on "Arithmetic" in the Encyclopædia Britannica, makes the following statement on this point:

After Archimedes, the most notable result was that given by Ptolemy, in the "Great Syntaxis." He made the ratio 3.141552, which was a very close approximation.

For several centuries there was little progress towards a more accurate determination of the ratio. Among the Hindoos, as early as the sixth century, the now well-known value, 3.1416, had been obtained by Arya-Bhata, and a little later another of their mathematicians came to the conclusion that the square root of 10 was the true value of the ratio. He was led to this by calculating the perimeters of the successive inscribed polygons of 12, 24, 48, and 96 sides, and finding that the greater the number of sides the nearer the perimeter of the polygon approached the square root of 10. He therefore thought that the perimeter or circumference of the circle itself would be the square root of exactly 10. It is too great, however, being 3.1622 instead of 3.14159... The same idea is attributed to Bovillus, by Montucla.

By calculating the perimeters of the inscribed and circumscribed polygons, Vieta (1579) carried his approximation to ten fractional places, and in 1585 Peter Metius, the father of Adrian, by a lucky step reached the now famous fraction 355⁄113, or 3.14159292, which is correct to the sixth fractional place. The error does not exceed one part in thirteen millions.

At the beginning of the seventeenth century, Ludolph Van Ceulen reached 35 places. This result, which "in his life he found by much labor," was engraved upon his tombstone in St. Peter's Church, Leyden. The monument has now unfortunately disappeared.

From this time on, various mathematicians succeeded, by improved methods, in increasing the approximation. Thus in 1705, Abraham Sharp carried it to 72 places; Machin (1706) to 100 places; Rutherford (1841) to 208 places, and Mr. Shanks in 1853, to 607 places. The same computer in 1873 reached the enormous number of 707 places.

Printed in type of the same size as that used on this page, these figures would form a line nearly six feet long.

As a matter of interest I give here the value of the ratio of the circumference to the diameter, to 127 places:

The degree of accuracy which may be attained by using a ratio carried to only ten fractional places, far exceeds anything that can be required in even the finest work, and indeed it is beyond anything attainable by means of our present tools and instruments. For example: If the length of a curve of 100 feet radius were determined by a value of ten fractional places, the result would not err by the one-millionth part of an inch, a quantity which is quite invisible under the best microscopes of the present day. This shows us that in any calculations relating to the dimensions of the earth, such as longitude, etc., we have at our command, in the 127 places of figures given above, an exactness which for all practical purposes may be regarded as absolute. This will be best appreciated by a consideration of the fact that if the earth were a perfect sphere and if we knew its exact diameter, we could calculate so exactly the length of an iron hoop which would go round it, that the difference produced by a change of temperature equal to the millionth of a millionth part of a degree Fahrenheit, would far exceed the error arising from the difference between the true ratio and the result thus reached.

Such minute quantities are far beyond the powers of conception of even the most thoroughly trained human mind, but when we come to use six and seven hundred places the results are simply astounding. Professor De Morgan, in his "Budget of Paradoxes," gives the following illustration of the extreme accuracy which might be attained by the use of 607 fractional places, the highest number which had been reached when he wrote:

It would of course be impossible for any human mind to grasp the range of such an illustration as that just given. At the same time these illustrations do serve in some measure to give us an impression, if not an idea, of the vastness on the one hand and the minuteness on the other of the measurements with which we are dealing. I therefore offer no apology for giving another example of the nearness to absolute accuracy with which the circle has been "squared."

It is common knowledge that light travels with a velocity of about 185,000 miles per second. In other words, light would go completely round the earth in a little more than one-eighth of a second, or, as Herschel puts it, in less time than it would take a swift runner to make a single stride. Taking this distance of 185,000 miles per second as our unit of measurement, let us apply it as follows:

It is generally believed that our solar system is but an individual unit in a stellar system which may include hundreds of thousands of suns like our own, with all their attendant planets and moons. This stellar system again may be to some higher system what our solar system is to our own stellar system, and there may be several such gradations of systems, all going to form one complete whole which, for want of a better name, I shall call a universe. Now this universe, complete in itself, may be finite and separated from all other systems of a similar kind by an empty space, across which even gravitation cannot exert its influence. Let us suppose that the imaginary boundary of this great universe is a perfect circle, the