An early American silversmith and active voice of the Revolution, Paul Revere is most well known for his ride to Lexington to alert the militia of the movement of the British troops and to warn John Hancock and Samuel Adams of their pending arrest. After his business fell into financial ruin after the implementation of the Intolerable Acts, he became one of the main organizers of the intelligence and alarm system which would keep tabs on the British military.

Despite popular historical accounts, he never rode hundreds of miles through New England shouting "the British are coming," but his acts in secret, and in public, did do a great deal to promote the American Revolutionary cause. Although little is known about William Mackay, history has preserved the fact that he was a Bostonian merchant strongly affected by the implementation of the Stamp Act of , and worked as a Son of Liberty to promote the Revolution.

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A lawyer and member of the Massachusetts provincial assembly, James Otis, Jr. He is famous for coining the phrase, "taxation without representation is tyranny," which would go on to become an anthem for patriot opposition to the crown. As a lawyer, Otis became enraged by British imposed "writs of assistance" which permitted British authorities to enter the house of a colonist without notice or probable cause. In response these writs, Otis gave a number of speeches speaking out against British exploitation of the colonists, and his words inspired many people to rise up against the tyranny of the crown.

The founder of Dickinson college in Pennsylvania, signatory to the Declaration of Independence , and Founding Father of the United States, Benjamin Rush is certainly one of the more famous voices of the revolution. Incredibly outspoken, Rush would make many friends, and enemies most notably George Washington, to whom he gave harsh criticism , and his words would effect great movements towards opposition to Great Britain.

Also, Thomas Paine looked to Rush when he was drafting his treatises in favor of the Revolutionary War. A truly learned manRush would join the Medical Committee of the Continental Congress, taking on a more practical role as well as a political one. Nicknamed "King Sears" for his pivotal role in organizing the New York mob, Isaac Sears was a leading member of the Sons of Liberty who leaned predominately towards orchestrating violence and encouraging anti-British demonstration.

A prosperous New York City merchant outraged by the Intolerable Acts, Sears was forceful in his opposition to the Stamp Act in particular, using whatever means necessary to dissuade the use of British stamps in the colonies. After the Stamp Act was finally repealed, he erected a number of liberty poles and broadsides large sheets of paper printed on one side only , signed "the Mohawks," warning that action would be taken against anyone supporting any of the Intolerable Acts.

He would eventually become the stand in commander of New York City until Washington arrived to relieve him in Sympathetic to the American cause, Solomon joined the New York branch of the Sons of Liberty and was arrested as a spy in After eighteen months of torture aboard the British vessel, he was released under the stipulation that he would remain as an interpreter for British-commissioned mercenaries.

While involved in this forced employment, Solomon helped many American prisoners escape their confinement and encouraged the mercenaries to join the Americans. Born in Scotland, James Swan moved to American colonies in the late s where he spent his youth as a shop clerk in Boston. As time went by, he became increasingly interested in the American Revolutionary effort and joined leagues with the Sons of Liberty. As a writer, he published many tracts and articles in opposition to the British crown.

Wealthy before the war, he financed Revolutionary efforts until he came to ruin in the early s. As secretary to the Continental Congress for fifteen years, Thomson was able to be directly involved in foreign affairs. Yet, he was a fiery individual who had many enemies. Famously, one James Searle attacked Thomson on the floor of Congress over a supposed misquotation, and the ensuing cane fight ended with both men being cut in the face.

## An International Quarterly

As consequence for his courage, the British made an example of Young, punishing him to such severity that he nearly died. Equal parts military man and cabinet maker, Marinus Willett had a reputation for street brawling and reckless behavior. He became a leading member of the New York faction of the Sons of Liberty, organizing surprise movements against the British. Most notably, in , he assembled a small band of men, commandeered a British sloop, and captured a protected British storehouse in Turtle Bay.

A signatory to the United States Declaration of Independence as well as the Articles of Confederation , Wolcott had a minor role in the Sons of Liberty and would go on to become the fourth Governor of Connecticut. One of his major acts for the Revolutionary cause was in erecting a shed on his country estate in Litchfield and, with help from his neighbors, casted more than 40, bullets. Call to order: or order pocket constitution books online. All rights reserved. Oak Hill Publishing Company. Box , Naperville, IL For questions or comments about this site please email us at info constitutionfacts.

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### By Flavius Josephus

Joseph Warren. Paul Revere. Alexander McDougall. When Newton at first believed the great comet observed November —March to be a pair of comets moving as Kepler proposed in straight lines, although in opposite directions, it was Flamsteed who convinced him that there was only one, observed coming and going, and that it must have turned about the sun.

Such a parabolic path had been shown in book I to result from the inverse-square law under certain initial conditions, differing from those producing ellipses and hyperbolas. In , Halley postulated that the path of the comet of was an elongated ellipse—a path not very distinguishable from a parabola in the region of the sun, but significantly different in that the ellipse implies periodic returns of the cornet—and worked out the details with Newton.

It is too often described as a treatise in the style of Greek geometry, since on superficial examination it appears to have been written in a synthetic geometrical style. The doctrine of limits occurs in the Principia in a set of eleven lemmas that constitute section 1 of book I. I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios; and so to premise, as short as I could, the demonstrations of those limits.

For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with greater safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curved lines for right ones, I would not be understood to mean indivisibles, but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas.

Section 1 of book I is unambiguous in its statement that the treatise to follow is based on theorems of which the truth and demonstration almost always depend on the taking of limits. Of course, the occasional analytical intrusions in book I and the explicit use of the fluxional method in book II notably in section 2 show the mathematical character of the book as a whole, as does the occasional but characteristic introduction of the methods of expansion in infinite series.

A careful reading of almost any proof in book I will, moreover, demonstrate the truly limital or infinitesimal character of the work as a whole. But nowhere in the Principia or in any other generally accessible manuscript did Newton write any of the equations of dynamics as fluxions, as Maciaurin did later on. The similarity of section 1, book I, to the introductory portion of the later De quadratura should not be taken to mean that in the Principia Newton developed his principles of natural philosophy on the basis of first and last ratios exclusively, since in the Principia Newton presented not one, but rather three modes of presentation of his fluxional or infinitesimal calculus.

A second approach to the calculus occurs in section 2, book 11, notably in lemma 2, in which Newton introduced the concept and method of moments. This represents the first printed statement in the first edition of by Newton himself of his new mathematics, apart from its application to physics with which the opening discussion of limits in section 1, book I is concerned.

This method I have interwoven with that other of working in equations, by reducing them to infinite series. The diminished area is. And, in general, the moment of A n is shown to be naA n-1 for n as a positive integer. We are to conceive them as the just nascent principles of finite magnitudes.

Newton thus offered in the Principia three modes of interpretation of the new analysis: that in terms of infinitesimals used in his De analysi … ; that in terms of prime and ultimate ratios or limits given particularly in De quadratura, and the view which he seems to have considered most rigorous ; and that in terms of fluxions given in his Methodus fluxionum , and one which appears to have appealed most strongly to his imagination.

From the point of view of mathematics, proposition 10, book II, may particularly attract our attention. Here Newton boldly displayed his methods of using the terms of a converging series to solve problems and his method of second differences. The proposition is of particular interest for at least two reasons. First, its proof and exposition or exemplification are highly analytic and not geometric or synthetic , as are most proofs in the Principia.

As a result, Newton had Cotes reprint a whole signature and an additional leaf of the already printed text of the second edition; these pages thus appear as cancels in every copy of this edition of the Principia that has been recorded.

## THE ANTIQUITIES OF THE JEWS

From at least onward, Newton attempted to impose upon the Principia a mode of composition that could lend support to his position in the priority dispute with Leibniz: he wished to demonstrate that he had actually composed the Principia by analysis and had rewritten the work synthetically.

He affirmed this claim, in and after , in several manuscript versions of prefaces to planned new editions of the Principia both with or without De quadratura as a supplement. It is indeed plausible to argue that much of the Principia was based upon an infinitesimal analysis, veiled by the traditional form of Greek synthetic geometry, but the question remains whether Newton drew upon working papers in which in extreme form he gave solutions in dotted fluxions to problems that he later presented geometrically.

But, additionally, there is no evidence that Newton used an analytic method of ordinary fluxional form to discover the propositions he presented synthetically. All evidence indicates that Newton had actually found the propositions in the Principia in essentially the way in which he there presented them to his readers. He did, however, use algebraic methods to determine the solid of least resistance. But in this case, he did not make the discovery by analysis and then recast it as an example of synthesis; he simply stated his result without proof.

It has already been mentioned that Newton did make explicit use of the infinitesimal calculus in section 2, book II, of the Principia , and that in that work he often employed his favored method of infinite series. And proposition 41 of book I is, moreover, an obvious exercise in the calculus. Newton himself never did bring out an edition of the Principia together with a version of De quadratura.

His strategy was to develop the subject of general dynamics from a mathematical point of view in book I, then to apply his most important results to solving astronomical and physical problems in book III. Book I opens with a series of definitions and axioms, followed by a set of mathematical principles and procedural rules for the use of limits; book III begins with general precepts concerning empirical science and a presentation of the phenomenological bases of celestial mechanics, based on observation.

It is clear to any careful reader that Newton was, in book I, developing mathematical principles of motion chiefly so that he might apply them to the physical conditions of experiment and observation in book III, on the system of the world. In fact, in book I as in book II , he tended to follow his inspiration to whatever aspect of any topic might prove of mathematical interest, often going far beyond any possible physical application.

While this mode of presentation makes the Principia more difficult for the reader, it does have the decided advantage of separating the Newtonian principles as they apply to the physical universe from the details of the mathematics from which they derive. I [of book III], and Prop.

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But the main hurdle for any would-be student of the treatise lies elsewhere, in the essential mathematical difficulty of the main subject matter, celestial mechanics, however presented. Even so, dynamics was taught directly from the Principia at Cambridge until well into the twentieth century. Nevertheless, the work might have been easier to read today had Newton chosen to rely to a greater extent on general algorithms.

In other words, these statements were to be regarded as not necessarily true, but only contingently phenomenologically so.

He also showed that the rate of free fall of bodies is not constant, as Galileo had supposed, but varies with distance from the center of the earth and with latitude along the surface of the earth.