NEWTON'S METHOD OF APPROXIMATION.

516. Find two numbers, one greater and the other less than a root of the equation (Art. 474). Let a be one of those numbers, the nearest to the root, if it can be ascertained. Substitute ay for x in the given equation; then y is small, and by omitting y3, y3, . . . . ., as in Horner's method (Art. 487), a value of y is obtained, which, added to a, gives b, a closer approximation to the value of x. Now substitute bz for x in the given equation, and a second approximation may be obtained by the same process as before. By proceeding in this way, the value of the root may be obtained to any required degree of accuracy.

The assumed value of x should be nearer to one root than to any other, in order to secure accuracy in the approximation.

EXAMPLES.

1. Find the real root of the equation 3

- 2 x

50.

When 2 and 3 are substituted for x in the equation, the results are 1 and 16, respectively; hence a root lies between 2 and 3,

and near to 2. Substitute 2+y for x, and there results

1 -- 10 y — 6 y3 — y3

whence, approximately, y .1.

=

=

0,

Now substitute 2.1 +z for x, and there results

Find one root of each of the following equations.

“NEW, AND UP WITH THE TIMES."

GREENLEAF'S

New Mathematical Șeries.

MERITORIOUS AND ECONOMICAL, Tested and proved, these books have an unprecedented popularity. Over 1,600,000 Copies in use!

DISTINGUISHING FEATURES.

I. A complete course for all classes of learners, upon the same plan and by one author.

2. The Primary Books are strictly elementary, and not mere abstracts or abridgments of larger works.

3. Each book is complete in itself, but so constructed as to prepare the learner for systematic progress.

4. The methods and processes are entirely modern, and most happily combine mental and written processes.

5. The course is arranged with a view of saving the time of the learner, and of expense, by the use of the smallest number of text-books.

6. The several books are beautifully illustrated, printed upon sized paper, and handsomely and durably bound.

SCHOOL OFFICERS

Contemplating a change of Mathematical Text-Books, are invited to examine GREENLEAF'S NEW SERIES. Address,

ROBERT S. DAVIS & CO.,

36 Bromfield St., Boston.

་

GREENLEAF'S NEW MATHEMATICAL SERIES.

GREENLEAF'S

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Each Book is complete in itself, and sold separately.

This book is on the plan of OBJECT TEACHING, and is beautifully illustrated.

It combines WRItten and Oral EXERCISES, in lessons, all of an inviting and attractive nature.

PRIMARY CLASSES are fully prepared by its use for the next book of the

series.

NEW ELEMENTARY ARITHMETIC.

A COMPREHENSIVE MANUAL, at once inductive, analytic and useful, expressly prepared to provide for new educational demands.

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A COMPLETE ARITHMETIC, entirely modern in its methods, and combining throughout practical utility with scientific accuracy.

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GREENLEAF'S is the standard in upwards of 1,000 CITIES and TOWNS in the NEW ENGLAND STATES alone.

TESTIMONIALS.

From REV. LEVERETT GRIGGS, School Visitor, Bristol, Conn., April 3, 1875.-"Some years since, Greenleaf's Arithmetics were used in our schools. They were displaced by

Last Fall the Board unanimously adopted Greenleaf's New Series, to the great satisfaction of our most accomplished teachers, as well as that of our school officers."

From S. B. FROST, Principal of Union Graded School, Killingly, Conn., April 29, 1876. "Greenleaf's New Series of Arithmetics were recently adopted for the Schools of Killingly. Teachers and pupils seem more than pleased with them. The series is well graded through its successive books; admirably arranged, concise and logical in definitions, and seems to present al the new methods which recent improvements made in teaching have developed. The New Practical Arithmetic is well named. It is sufficiently full for all ordinary purposes, and is characterized by a happy combination of beauty in the type, clearness and preciseness in the language, and is, I think, the best work of the kind extant."

From LINUS T. GILBERT, School Visitor, Litchfield, Conn., Jan. 2, 1875.-" Greenleaf's New Series of Arithmetics are just what we have wanted for some time. I think them far superior formerly in use. I am pleased that they have been introduced in our schools. There are many new features in the books that must commend them to towns now using books like as we formerly had."

to

From S. T. DUTTON, Principal South Norwalk School, Ct., April, 1875.-"I have used Greenleaf's Mathematical Works, and take pleasure in testifying to the rare excellencies which they possess. So well are they adapted to the popular wants that I never heard a teacher speak of them as deficient in any respect."