those parts, for every ten minutes of the quadrant, by the decimal notation. This work was further prosecuted by Regiomontanus, the disciple and friend of Purbach; but as the plan of his master was evidently defective, he afterwards changed it altogether, by computing anew the table of sines for every minute of the quadrant, to the radius 1000000. Regiomontanus also introduced the use of tangents into Trigonometry, the table of which he named Canon Facundus, on account of the numerous advantages arising from its use. He likewise enriched the science with many valuable theorems and precepts; so that, excepting the use of logarithms, the Trigonometry of Regiomontanus was little inferior to that of our own times.

About this period the mathematical sciences began to be studied with ardour in several parts of Italy and Germany, and it can hardly be supposed that a science so obviously useful as Trigonometry, would be without its share of admirers and cultivators, although scarcely any of their writings on the subject have been committed to the press. John Werner of Nuremburg, (who was born in 1468, and died in 1588,) is said to have written five books on triangles; but whether the work exists at present, or is lost, we are not informed. A brief treatise on plane and spherical Trigonometry was written about the year 1500, by Nicholas Copernicus, the celebrated restorer of the true solar system. This tract contains the description and construction of the canon of chords, nearly in the manner of Ptolemy; to which is subjoined a table of sines to the radius 100000 with their differences, for every ten minutes of the quadrant, the whole forming a part of the first book of his Revolutiones Orbium Cœlestium, first published at Nuremburg, fol. 1543. Ten years after, Erasmus Reinhold, Professor of Mathematics at Wirtemburg, published his Canon Facundus, or table of tangents; and about the same

time Franciscus Maurolycus, Abbot of Messina, in Sicily, and one of the best Geometers of the age, published his Tabula Benefica, or canon of secants.

But a more complete work on the subject than any that had hitherto appeared, was a treatise in two parts by Vieta one of the ablest mathematicians in Europe, published at Paris, in 1579. The first part, entitled Canon Mathemati cus seu ad triangula, cum appendicibus, contains a table of sines, tangents, and secants, with their differences for every minute of the quadrant, to the radius 100000. The tangents and secants towards the end of the quadrant are carried to 8 or 9 figures, and the arrangement is similar to that at present in use, each number and its compliment standing in the same line, opposite one another. The second part of this volume, entitled Universalium Inspectionum ad Canonem Mathematicum liber singularis, contains the construction of the foregoing table, a complete treatise on plain and spherical Trigonometry, with their application to various parts of the Mathematics; particulars relating to the quadrature of the circle, the duplication of the cube; with a variety of other curious and interesting problems and observations of a miscellaneous nature. Besides the above masterly performance, Vieta was the author of several tracts on plane and spherical Trigonometry, which may be found in the collection of his works, published by Schooten, at Leyden, in 1646.

The triangular canon was next undertaken by George Joachim Rheticus, a pupil of the great Copernicus, and Professor of Mathematics at Wirtemburg; "he computed the

f For further particulars of this interesting volume, see The History of Trigonometrical Tables, p. 4, 5, 6, 7, by Dr. Hutton. It appears that scarcely any copies of this excellent work are now to be found; for the Doctor says, in concluding his account of it, "I never saw one (copy) besides that which is in my own possession, nor ever met with any other person at all acquainted with such a book," p. 7.

anon of sines and co-sines for every ten seconds of the uadrant, and for every single second of the first and last egree;" he had proposed, in obedience to the desire of his master, to complete the trigonometrical canon, and extend further than had hitherto been done; but, dying in 1576, the completion of this vast design was at his request consigned to his pupil and friend Valentine Otho, mathematician to the Electoral Prince Palatine; who, after everal years of indefetigable labour and intense application, accomplished the work, and it was printed at Heidelberg, in 1596, under the title of Opus Palatinum de Triangulis. We have here an entire table of sines, tangents, and secants, for every ten seconds of the quadrant to ten places of figures, with their differences, being the first complete canon of these numbers that was ever published.

But notwithstanding the pains that had been taken in the calculation, the tables in this valuable performance were afterwards found to contain a considerable number of errors, particularly in the co-tangents and co-secants; the correction of these was undertaken by Bartholomew Pitiscus, a skilful mathematician of that time, who, having procured the original manuscript of Rheticus, added to it an auxiliary table of sines to 21 places, for the purpose of supplying the defect of the former, and published both in folio, at Frankfort, in 1613, under the title of Thesaurus Mathematicus, &c. Pitiscus then re-calculated the co-tangents and co-secants to the end of the first six degrees in Otho's work, which rendered it sufficiently exact for astronomical purposes, and published his corrections in separate sheets, making in the whole 86 pages in folio.

The Geometrica Triangulorum of Philip Lansbergius, in four books, was published in 1591; a brief, but very elegant work, containing the canon of sines, tangents, and secants, with their construction and application in the solution of

2. The sides about one of the angles of the base of a rectangular prism are 7 and 5 respectively, and the altitude of the prism 20; required the solidity?

Thus 7x5=35 area of the base; then 35 × 20=700 the solidity.

3. The sides of the base of a triangular prism are 2, 3, and 4, respectively, and the perpendicular altitude 30; required the solidity?

Thus (Art. 284.) p=

2+3+4
2

=4.5, and

/4.5×4.5—2×4.5-3×4.5—4=√√/8.4375=2.9047375=area

of the base.

Then 2.9047375 × 30=87.1421250=the solidity.

4. The base of a prism is a regular hexagon, the side of which is 8 inches, and the altitude of the prism is 4 feet; required the solidity?

Here (Art. 285.) b=8, a=√/8° -4° = {√/4S=) 6.9282, nba 6 × 8 × 6.9282 n==6, and =

2

2

-=166.2768 square inches the

area of the base: wherefore by the rule 166.2768 × 48 (inches) =7981.2864 cubic inches=4 cubic feet 1069.2864 cubic inches..

5. The length of a parallelopiped is 16 feet, its breadth 4 feet, and thickness 6 feet; required the solidity? Ans. 486 cubic feet.

6. The length of a prism is 5 feet, and its base an equilateral triangle, the side of which is 24 feet; required the solidity? Ans. 13.5315 cubic feet.

7. The base is a regular pentagon, the side of which is 12 inches, and the length 9 feet; required the solidity of the prism?

291. To find the solid content of a pyramid.

RULE. Find the solid content of a prism, having the same base and altitude as the pyramid, by the last rule; one third part of this prism will be the solid content of the pyramid e.

EXAMPLES.-1. The altitude of a pyramid is 20 feet, and its base is a square, the side of which is 12 feet; required the solidity?

• This depends on cor, 1. 7. 12. Euclid.

Here (Art. 282.) 12 × 12=144 area of the base, 144 × 20 =2880 solidity of the circumscribing prism, and -=960

=

cubic feet the solid content of the pyramid.

2880
3

2. The altitude of a pyramid is 11 feet, and its base a regular hexagon, the side of which is 4 feet; what is the solidity?

Here (Art. 285.) b=4, a=√/42—22= √√/12=3.4641016,
nba 6 x 4 x 3.4641016
41.5692192-area of the

n=6, and

2

2

base; also 41.5692192 × 11=457.2614112=solidity of the cir

457.2614112

cumscribing prism (Art. 290.), ...

3

152.4204704

=

cubic feet the solidity of the pyramid.

3. What is the solid content of a triangular pyramid, the height of which is 10, and each side of the base 3? Answer, 12.99039.

4. What is the solidity of a square pyramid, each side of its base being 13, and the altitude 25 ?

292. To find the solid content of a cylinder.

RULE. Multiply the area of the base by the perpendicular altitude, and the product will be the solidity f.

f This rule depends on Euclid 11 and 14 of book 12. The convex superficies of a cylinder is found by multiplying the circumference of the base by the altitude of the cylinder; to which, if the areas of the two ends be added, the sum will be the whole external superficies.

To find the solidity of squared timber. 1. If the stick be equally broad and thick throughout, find the area of a section any where taken, and multiply it into the length, the product will be the solidity. 2. If the stick tapers regularly from one end to the other, find half the sum of the areas of the two ends, and multiply it into the length. 3. If the stick does not taper regularly, find the areas of several different sections, add them together, and divide the sum by the number of sections taken, this quotient multiplied into the length, will give the solidity.

To find the solidity of rough or unsquared timber. Multiply the square of one fifth of the mean girt by twice the length, and the product will be the solidity. Or, multiply the square of the circumference by the length, take of the product, and from this last number subtract of itself, the remainder will be the solidity. See on this subject Hutton's and Bonnycastle's Treatises on Mensuration.