THE present Publication being chiefly intended for the American reader, we have taken the liberty of making some alterations, which, it is presumed, will rather prove advantageous than otherwise.

Some parts of the work have been abridged, and other parts totally omitted, as being of little or no use to the American Surveyor. These alterations have enabled the Editor to introduce into the body of the work, some matters of considerable importance in the practice of surveying; amongst which, are a complete sct of tables of latitude and departure, to the distance of 100, and to every 15 Minutes of the quadrant; a table of logarithms from 1 to 10,000; and a table of artificial sines, tangents, and secants ; also an example of calculating the contents of a survey, according to the method commonly practised in the surveyor general's office of this commonwealth,

Philadelphia, 1808.

4. Men

VYSE'S TUTOR'S GUIDE, BEING a complete System of trithmetic; with vous Branches in the Mathematics.

In fix Part., viz 1. Arithmetic in all its useful Rules, and a great variety of qu-stions. 2 Vulgar Fractions in all their Parts.

3 Decii si fractions, with the Extraction of Roots of different Powers; to which is added, Rules, &c. for the easy Calculation of interest and innuities, &c furation of Supertices and Solids, applied to measuring Artificers' Works, &c. with a collection of Questions for Exercise. To which is added, the Specific gravity of Me. tals, &c. 5. Chronology or the Method of finding the several Cycles, Epacts. Moveables Feafts, Time of High Water, &c.; with a collection of Questions relating to History; likewise all the moft useful Examples on both the Globes. 6. Algebra, wherein the Method of raising and resolving Equations is rendered casy, and illustrated with a Variety of Examples and Numerical Questions.

THE KEY TO THE TUTOR'S GUIDE; OR the Arithmetician's Repository : containing Solutions of the Questions, &c. in the Tutor's Guide. The whole being principally designed for the Ease of Schoolmasters; and, with the Guide, furnishes a more complete and extensive System of Arithmetic than any extant ; and will enable all those who are acquainted with the first Principles to attain a competent Knowledge of the several Rules with Ease and Precision, and thereby qualify them. selves to instruct the children and domestics of a family at leisure hours.

AN INTRODUCTION TO ALGEBRA; WITH notes and observations ; designed for the use of Schools and Places of Public Education. by John Bong NYCASTLE, of the Royal Military Academy, Woolwich. The first American edition.

The American Tutor's Asistant ; or, a compendious system of Practical Arithmetic ; containing, the several rules of that ufeful science, concisely defined, methodical. ly arranged, and fully exemplified. The whole particularly adapted to the case and regular instruction of youth. in ov: American schools.





HE word geometry imports no

than to measure the earth, or to measure land; yet in a larger and more proper sense, it is applied to all sorts of dimensi

It is generally supposed to have had its rise among the Egyptians, from the river Nile's 'destroying and confounding all their land-marks, by its annual inundations, which laid them under the necessity of inventing certain methods and measures to enable them to distinguish and adjust the limits of their respective grounds, when the waters were withdrawn. And this opinion is not entirely to be rejected, when we consider that Moses is said to have acquired this art when he resided at the Egyptian court. ' And Achilles Tatius in the beginning of his introduction to Aratus's Phapomena, informs us, that the Egyptians were the first who measured the heavens and the earth (and of course the earth first) and


that their science in this matter, was engraven on columns, and by that means delivered to posterity.



It is a

matter of some wonder, that though surveying appears to have been the first, or at least one of the first of the mathematical sciences, that the rest have met with much greater improvements from the pens of the most eminent mathematicians, while this seems to have been neglected ; insomuch that I have not been able to meet with one author, who has sufficiently explained the whole art in its theory and practice: for the most part, it has been treated of in a practical manner only; and the few who have undertaken the theory, have in a great measure omitted the practice.

These considerations induced ine to attempt a methodical, easy, and clear course of Surveying ; how far I have succeeded in it, must be deterniined by the impartial reader: the steps I have taken to render the whole evident and fainiliar are as follow :

In section the first, you have decimal fractions, the square root, geometrical definitions, some necessary theorems and problems; with the nature and use of the tables of logarithm numbers, sines, tangents, and



The second section contains plane trigonometry right angled and oblique, with its application in determining the measures of inaccessible heights and distances,

The third section gives an account of the chains and measures used in Great-Britain an Ireland; methods of surveying and of taking inaccessible distances by the chain only, with some necessary problems; also a particular description of the several instruments used in surveying, with their respective uses.

The fourth section contains two methods of finding the areas of maps

from their ge ometrical construction,

more concise · than any heretofore made public.

The fifth section contains a new, and much more concise method of determining the areas of surveys from the field-notes, or by calculation than any hitherto published; and I venture to assert that it is impossible (from the nature of right-lined figures) that any method or methods more concise than this, can be investigated.

To these methods is annexed a short table of difference of latitude and half departure, to every degree and quarter of a degree of the quadrant, the stationary distance being


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