The science of the earth and the history of man, though cultivated by very different classes of specialists and in very different ways, must have their meeting-place. They must indeed not only meet, but overlap and run abreast of each other throughout nearly the whole time occupied by the existence of man on the earth. The geologist, from his point of view, studies all the stratified crust of the earth, down to the mud deposited by last year's river inundations. The historian, aided by the archæologist, has written and monumental evidence carrying him back to the time of the earliest known men, many thousands of years ago. Throughout all this interval the two records must have run more or less parallel to each other, and must be in contact along the whole line. The geologist, ascending from the oldest and lowest portions of the earth's crust, and dealing for millions of years with physical forces and the instinctive powers of animals alone, at length as he approaches the surface finds himself in contact with an entirely new agency, the free-will and conscious action of man. It is true that at first the effects of these are small, and the time in which they have been active is insignificant in comparison with that occupied by previous geological ages; but they introduce new questions which constantly grow in importance, down to those later times in which human agency has so profoundly affected the surface of the earth and its living inhabitants. Finally, the geologist is obliged to have recourse to human observation and testimony for his information respecting those modern causes to which he has to appeal for the explanation of former changes, and has to adduce effects produced by human agency in illustration of, or in contrast with, mutations in the pre-human periods. The historian, on the other hand, finds, as he passes backward into earlier ages, documentary evidence failing him, and much of what he can obtain becoming mythical, vague or uncertain, or difficult of explanation by modern analogies, until at length he is fain to have recourse to the pick-axe and spade, and to endeavour to disinter from the earth the scanty relics of primeval man, much as the geologist searches in the bedded rocks for the fossils which they contain. He has even learned to use for these earliest ages the term prehistoric, and so practically to transfer them to the domain of the archæologist and geologist.

In the first full-length biography of John William Dawson (1820-1899), eminent scientist and principal of McGill University, Susan Sheets-Pyenson highlights the extraordinary scope of Dawson's educational and scientific career and his commitment to science, rationality, and the advancement of knowledge.

John Bridge (d.1665), a widower with two sons, emigrated in 1631 from England to Cambridge, Massachusetts. In 1658 he married widow Eliza- beth Saunders, widow of Martin Saunders and earlier widow of Roger Bancroft; they had no children, and she married again after John's death. Descendants and relatives lived in New England, New York, New Jersey, Ohio, Indiana, Michigan, Illinois, Wisconsin, Minnesota and elsewhere. Includes some ancestry and relatives in England.

This monograph considers several well-known mathematical theorems and asks the question, “Why prove it again?” while examining alternative proofs. It explores the different rationales mathematicians may have for pursuing and presenting new proofs of previously established results, as well as how they judge whether two proofs of a given result are different. While a number of books have examined alternative proofs of individual theorems, this is the first that presents comparative case studies of other methods for a variety of different theorems. The author begins by laying out the criteria for distinguishing among proofs and enumerates reasons why new proofs have, for so long, played a prominent role in mathematical practice. He then outlines various purposes that alternative proofs may serve. Each chapter that follows provides a detailed case study of alternative proofs for particular theorems, including the Pythagorean Theorem, the Fundamental Theorem of Arithmetic, Desargues’ Theorem, the Prime Number Theorem, and the proof of the irreducibility of cyclotomic polynomials. Why Prove It Again? will appeal to a broad range of readers, including historians and philosophers of mathematics, students, and practicing mathematicians. Additionally, teachers will find it to be a useful source of alternative methods of presenting material to their students.