MADE PLAIN AND EASY TO THOSE WHO HAVE NOT STUDIED MATHEMATICS. CONTAINING THE ELEMENTARY PRINCIPLES OF THE SCIENCE; ALL THE RULES AND TABLES NECESSARY FOR MAKING ALL THE CALCULATIONS FOR AN ALMANAC; And for the solution of more than one hundred Astronomical and Chronological Problems; with sketches of Geometry, Trigonometry, Surveying, Levelling, and other branches of Mathematics, depending on the principles of Geometry: also tha construction and use of Logarithms a Table of the Logarithms of Numbers, of Logarithmic Sines, Tangents, and Secanis, and of Logistical Logarithms; and an Explanation of the manner of constructing the Tables, &c. &c. BY THOMAS SPOFFORD. [COPY-RIGHT SECURED ACCORDING TO LAW.J BOSTON-Published by LEMUEL GULLIVER, at STATIONERS' HALL, 82 State Street. NEW YORK:-By CHARLES SMALL, BOOKSELLER, No. 298 Pearl Street. 1835. ALL unacquainted with mathematics, who have wished to pursue the study of Astronomy, have always experienced the want of a concise work, containing the elementary principles of the science, with the rules and tables necessary for solving its most useful and interesting Problems. It was this that induced the author, a long time ago, to undertake the following: but it would not even now be presented to the public, were several of the late publications on the same subject, what their titles and prefases purport. With the exception of Gummere's excellent treatise, (to understand which, a thorough knowledge of the most abstruse branches of mathematics is required,) there has no work been produced in America, containing the precepts and TABLES necessary for making the calculations of a common Almanac. "Tis true, within a few years, several systems, purporting to be such, have made their appearance; but they are only compilations from Ferguson and others, containing little or nothing new, and all the radical ERRORS in the TABLES of those excellent and now obsolete works. A knowledge of Algebra, Conic Sections, and several other branches of mathematics, has generally been considered indispensable for any one who would know any thing of Astronomy: but the following Treatise is designed to render the science perfectly intelligible, and easy to be understood, by those who have no further knowledge of mathematics, than the simple rules of Arithmetic; and it is believed that the problems may all be readily solved by any person of common intellect, without the aid of a teacher. Many of the rules and tables are entirely new; those for the CALCULATION of ECLIPSES save more than NINE TENTHS of the time and trouble required to solve the problem in the usual tedious and intricate way of calculating by parallaxes, and are sufficiently accurate for common purposes. For the last seven years, the author has used no other rules or tables in making the calculations for his Almanac, and for several of the most popular Registers and Almanacs in the United States and the British Colonies. By these rules and tables, he has calculated all the eclipses that will happen for ONE HUNDRED YEARS; which could hardly be done by the old method, (parallaxes,) in the lifetime of a man, should he live to that age. To render the work more useful, enough of Geometry, Trigonometry, and several other branches of mathe matics, is given, to enable the reader to solve understandingly the most important and useful Problems in those studies. As the principles of Geometry are general and abstract truths, and exist in nature, inde pendently of lines and figures, it is thought that the learner may acquire a far better, and more lasting idea of these principles, by reasoning abstractly, and then making his own diagrams for illustration and demonstration, than by having figures first presented to the eye. For this reason, the Geometrical definitions and propositions are given in general terms, and without reference to, or the aid of diagrams; which leads the learner so to pursue the study of Geometry, as to improve that mental faculty, which enables the mind to comprehend general propositions, and to pursue trains of thought disconnected with sensible objects. For the same reason, examples are omitted in several Problems; for if the learner is able by the rule alone to solve the Problem, without any visible illustration, he will be much more likely to remember the solution, than if it was illustrated by an example wrought out. But in difficult Problems, examples are wrought at length. The erroneous opinion which has hitherto prevailed, that none could learn Astronomy without previously studying a tedious course of mathematics, has undoubtedly prevented many from beginning a study, the want of which they have severely felt, when, in after life, they have been engaged in business wherein this science was necessary. That any art or science may be made of most service to mankind, it is requisite that many understand it; for the united strength of numbers will readily accomplish what far exceeds the limited ability of a few. If means could be devised to extend and increase the knowledge of Astronomy, the advantages resulting to the community would be immensely great. The discoveries and improvements in the science have of late been many and great; but a dissemination of astronomical knowledge has not been commensurate with these, Could one object more be obtained, (which at first might seem easy,) the means of ascertaining longitude with the same facility that latitude is determined, this single discovery would be of more value to mankind, than the richest gold mine in the world. The government of Great Britain has frequently paid premiums of 10,000 pounds sterling, for only improvements in the method of determining longitude; and a reward of 50,000 pounds sterling is now offered by the same government, to any person who shall obtain this grand desideratum in Nautical Astronomy. That a competent knowledge of Astronomy for all common practical purposes, may be obtained without any previous knowledge of mathematics, except Arithmetic, is designed to be proved by this Treatise. How far the author has succeeded, is left to the judgment of the cand.d. INTRODUCTION. THE Science of astronomy is of great antiquity; indeed, we may date its origin with that of agriculture, or society itself: but the arts and sciences, like kingdoms and states, have had their various changes and revolutions; sometimes shining with uncommon lustre, and at others involved in obscurity and barbarity. Astronomy, after having flourished for a considerable time, under the auspices of Pythagoras and his followers, was again neglected and obscured for many ages, so that the true system of the universe seems to have been entirely lost and forgotten, and whimsical and erroneous ones were substituted in its place. Instead of consulting the heavens, and collecting the history of nature, succeeding philosophers were ambitious of gratifying their own vanity, by inventing systems, which had no conformity to fact and experiment. Solid orbs and epicycles were multiplied to answer every appearance, till the universe had lost all its native beauty in their descriptions, and seemed again reduced to a chaos by their unhappy labours. The erroneous hypotheses or systems of most note, were those of Claudius Ptolemy and Tycho Brahe. The former placed the earth at rest in the centre of the universe, and supposed all the other heavenly bodies to revolve round it, from east to west, once in about 24 hours. The latter also placed the earth in the centre of the universe, and supposed all the heavenly bodies to revolve round it, and the planets, in addition, to revolve round the sun. These systems were severally supported, till the commencement of the sixteenth century, when Copernicus, a bold and original genius, adopted the Pythagorean, or true system of the universe, and published it to the world in the year 1530, with new and demonstrative arguments in its favour. Animated with daring enthusiasm, he laid his hands on the cycles and crystal orbs of Ptolemy, and dashed them to pieces. And, with the same noble frenzy, he took the unwieldly earth, and sent her far from the centre of the system, to move round the sun with the rest of the planets; so that, of all the celestial equipage, with which she had been formerly dignified, there only remained the moon to attend and accompany her in her journey. This system is particularly described as the Solar System. As Europe, however, was still immersed in barbarism and ignorance, it was a long time before it was accepted as the true system: but the discoveries of Kepler and Galileo, (by means of the telescope, which was the invention of the latter,) confirmed it; and it has since, by Sir Isaac Newton, been established upon such an immoveable and everlasting basis of mathematical demonstration, as can never be shaken, while the present frame of nature exists. With regard to some scripture expressions, which, if taken in their literal sense, seem to contradict some of the principles of this system, it may be observed, that men of sense, in all ages, when not treating of the sciences, have adapted their language to the capacity of their hearers; and in this manner, no doubt, those passages are to be understood. Moses and Joshua might have been acquainted with the motion of the earth; but if they had said that the sun was at rest in the firinament, and that the rising and setting of the heavenly bodies were caused by the revolution of the earth on its axis, the Israelites, who were unskilled in human science, would rather have considered them as impostors, than as leaders commissioned by the Almighty to conduct them from bondage and servitude to the promised land. The moon is known to be a dark body, shining only by reflecting the light of the sun: it is also known to be the smallest body visible in the heavens; but it would be idle to tell the common people this, for it appears larger and more luminous than any other, except the sun. Moses, therefore, calls the moon, as well as the sun, a Great Luminary, because he adapted his language to the ideas and conceptions of the people whom he addressed. We cannot suppose that the scriptures were intended to instruct in the sciences, yet many parts clearly prove that their authors were eminent in human learning, as well as divine. The writer, for instance, of the book of Job, whoever he might be, was a profound scholar and philosopher, as well as theologian. Nothing human can in any way compare with the majestic grandeur, beauty, and sublimity of the language of that book-with the exalted ideas and conceptions it gives of the Supreme Being. And it is not in Job only, but in every other part of the Bible, that the most magnificent descriptions of the divine nature, and the works of creation, are given with such majestic beauty and brevity, as it is in vain to look for in any human composition. To the noble science of astronomy we are indebted for what we know of the most wonderful works of the Deity. In the heavens it is, that the All-wise Creator has chiefly manifested his greatness and power. It is here that sovereign wisdom shines with the greatest lustre and majesty, and that the most sublime ideas of order and harmony reign. In this innumerable host of celestial orbs, all is magnificence, proportion, and regularity. Yet this grand spectacle is not continually exhibited to our view as an object of idle admiration; it is much more connected with the wants and conveniences of man. Unassisted by astronomy, even the least instructed is aware that he would be ignorant even of his own age; as without it, we should have no chronology, and no calendar. It is in the heavens that are found the means of arresting time in its eagle flight, of determining those interesting eras from which we date the most important events, and of regulating the seasons of the year. We are indebted to the science of astronomy for the means by which we now traverse the unfathomable ocean with so much skill and security, and determine at any time our situation on this trackless element. By the interposition of the heavens, immense deserts and vast and unknown countries are explored, and their treasures transported to other regions destitute of these resources; and by this means also, the most distant nations hold their correspondence. It is not only by the knowledge of astronomy that we learn by what means or laws the Almighty carries on and continues the wonderful order, connexion, and harmony, that pervades the planetary system; but our very faculties are enlarged, our minds are exalted above the |