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instruction have proved, as hundreds of individuals would bear testimony, that the theorem here given will save, at the lowest estimate, two-thirds of the labor ordinarily incurred by the rectangular method. A further advantage is that, dispensing with a large and faulty table altogether, it is far more accurate—the computations being executed by aid of the common logarithmic numbers, calculated with greater care and usually extending to six or seven deci. mal places, and the operation being so ordered that, without any additional labor beyond what is absolutely essential to an honest confidence in the result, all gross errors, if any exist, whether of the field or the tables, are detected, and if these have no existence, the smaller and unavoidable ones very much reduced.

It is hoped that this book, often requested by my pupils, will prove acceptable to the schools generally.

Of the third Part, embracing the mensuration of solids, spherical trigonometry, and navigation, time will permit us to say little more than that, by the method pursued, we have been enabled, within moderate limits, to give a fuller development of these subjects than is usually found in our elementary books.

The modification and extension of Napier's Rules demands, how ever, a brief historic notice. I demonstrated and extended these rules by showing : I. When A = 90°, 2. = 90°-Q, B. = 90°- B, C. = 90° - C,

sinb = cosac cos B. = tan C. tanc,
sinc = cosac cos Cc tanB. tanb,
sinac = cosb cosc tanB. tang.
sinB = cos C. cosb tanac tanc,
sin C.= cos B, cosc

tana, tanb:
II. When a = 90°, B, = 180° — B, A, = A - 90°, &c.,

sinB = cos A, cosb, = tanc, tan C.,
sin C= cos A, cosc, = tanb, tanB.,
sin A,=cos B, cosC, = tanb, tanc,,
sinb, = cosc, cos B. = tan A, tan C.,

since = cosb, cos C. = tan A, tanB.: III. When c= a, or the triangle is isosceles,

sinac = tan Atan(1B). sin A= tana, tan(+b), sin(+b) = cosa, cos(B).. sin (1 B) = cos A, cos(16).

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Having shown the above extension to Mr. Dascum Green, then a pupil, he returned soon after, saying that he had not only verified my forms, but had obtained better ones, and presented the modi. fications of Napier's Rules as I had extended them, substantially as they will be found in the text.

This rule, as now extended and modified, possesses a greater simplicity and symmetry, and will enable us, in spherical astronomy, frequently to dispense with complicated figures.

I have added a small set of tables, extending to seven decimal places, calculated to answer the wants of the student while pursuing the work, and to make him more ready in using tabular numbers, by compelling him to interpolate by second differences. Afterwards he will find it decidedly to his advantage to possess himself of the tables recently published by Professor Stanley.

In conclusion, the advantages which we have endeavored to se

cure are:

1o. A better connected and more progressive method of geometrizing, calculated to enable the student to go alone.

2o. A fuller, more varied, and available practice, by the introduction of more than four hundred exercises, arithmetical, demonstrative, and algebraical, so chosen as to be serviceable rather than amusing, and so arranged as greatly to aid in the acquisition of the theory.

3o. The bringing together of such a body of geometrical knowledge, theoretical and practical, as every individual, laying any claim to a respectable education or entering into active life, demands.

4°. The furnishing to those who may wish to proceed on in mathematical learning, of a stepping-stone to 'higher and more ex. tended works. How well we have accomplished our object it is not, of course,

We have endeavored to render the work as mechan. ically correct as possible, but, residing at a distance from the place of publication, we can hardly expect that it will be entirely free from typographical imperfections.

G. C. W. Genesee Wesleyan Seminary, June, 1848.

for us to say.

CONTENTS.

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