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First university in the United States is a status asserted by more than one U.S. university. In the U.S. there is no official definition of what entitles an institution to be considered a university versus a college, and the common understanding of "university" has evolved over time. The 1911 Encyclopædia Britannica tells the story of the gradual emergence of U.S. "universities" thus:
|“||In the United States the word university has been applied to institutions of the most diverse character, and it is only since 1880 or thereabouts that an effort has been seriously made to distinguish between collegiate and university instruction; nor has that effort yet completely succeeded. Harvard, William and Mary, and Yale . . . were organized . . . on the plans of the English colleges which constitute the universities of Oxford and Cambridge. Graduates of Harvard and Yale carried these British traditions to other places, and similar colleges grew up in New York, New Jersey, Pennsylvania, New Hampshire and Rhode Island.... Around or near these nuclei, during the course of the 19th century, one or more professional schools were frequently attached, and so the word university was naturally applied to a group of schools associated more or less closely with a central school or college. Harvard, for example, most comprehensive of all, has seventeen distinct departments, and Yale has almost as many. Columbia and Penn have a similar scope. In the latter part of the 19th century Yale, Columbia, Princeton and Brown, in recognition of their enlargement, formally changed their titles from colleges to universities.||”|
The issue is further confused by the fact that at time of founding of many of the institutions in question, the U.S. didn't exist as a sovereign nation. Moreover, questions of institutional continuity sometimes make it difficult to determine the true "age" of any institution.
Several universities claim to be the first university in the United States:
Harvard University calls itself "the oldest institution of higher learning in the United States" and this claim is rarely challenged. William & Mary calls itself "America's second-oldest college", acknowledging Harvard's claim but adding that William & Mary itself is the nation's oldest college in its "antecedents".
It is possible to disagree what year should be taken as Harvard's "real" founding date (Harvard uses the earliest possible one, 1636, when the institution was chartered by the Massachusetts Bay Colony). However, Harvard has operated since 1650 under the same corporation, the "President and Fellows of Harvard College"; it thus has an unbroken institutional history dating back to the mid seventeenth century (an official Harvard web page for the School of Engineering and Applied Sciences  claims, "Founded in 1636, Harvard is America's oldest university").
As a historical curiosity, a College of Henricopolis or University of Henrico, near Richmond, Virginia, was chartered in 1618 and construction begun, but the buildings were destroyed with the town during the Indian Massacre of 1622 and not re-erected. At times, the College of William and Mary itself claimed to be the nation's first college "in its antecedents" and technically this is true: the foundational concept of the institution was laid decades before Harvard's founding.
The founding date of the University of Pennsylvania is associated with more subjectivity and institutional debate than the more straightforward dates used by the eight other colonial era colleges. Harvard University uses as its founding date 1636, the year in which the legislature of the Massachusetts Bay Colony formally voted to budget funds for the creation of a college in Newtowne, later called Cambridge. The seven remaining colonial era colleges consider their founding dates to be the year in which they were first granted charters and thus became legal corporations.
Penn's claim as the first university in the United States is three-fold: the 1765 founding of the first medical school in America made Penn the first institution to offer both "undergraduate" and professional education; the 1779 charter made it the first American institution of higher learning to take the name of "University"; and existing colleges were established as seminaries.
The first charter for an institution of higher learning in Philadelphia was granted in 1755 to the College of Philadelphia, a new undertaking of the Academy of Philadelphia, which had previously taught only secondary students. In 1779, a charter was granted to a separate institution called the "University of the State of Pennsylvania" ) which in 1791 was merged with the College of Philadelphia and issued a new charter as the "University of Pennsylvania."
Despite the three charter dates of 1755, 1779 and 1791, the University used for more than a century the founding date of 1749, the year in which founder Benjamin Franklin first convened a board of trustees to organize the new institution. In 1899, the University's board of trustees voted to change the founding date by nine years to 1740, the year in which a group of Philadelphia citizens established a trust for a charity school requested by traveling evangelist George Whitefield. The hall erected would include space allocated to the charity school for local orphans, a practice Whitefield suggested in many cities as he toured up and the down the East Coast. The frame of the building was erected, but the citizens discovered that they lacked the funds to furnish the interior chapel or open the charity school.
The unfinished edifice lay vacant for roughly a decade until Franklin's nascent Academy of Philadelphia was looking for space to begin operations and purchased the still unused building in 1750. The Academy of Philadelphia operated a charity school for a few years and this brief period was the basis for the trustees' claims of institutional continuity to the earlier date, as the Academy had assumed the trust of the charity school for local orphans planned but not begun by the original fundraisers of the building.
Parenthetically, the University of Pennsylvania calls itself the fourth oldest institution of higher learning in the United States, comparing the legal charter dates of Princeton University (1746) and Columbia University (1754) with the 1740 date in which the trust had been established and fundraising had begun for the building it would ultimately purchase in 1750. Secondary instruction began at the Academy of Philadelphia in 1751 and undergraduate education began at the College of Philadelphia in 1756.
Perhaps not surprisingly, Princeton University and Columbia University do not follow the same train of thought in their own institutional histories. Princeton and Columbia consider themselves the fourth and fifth oldest institutions of higher learning in the country, respectively, comparing the three collegiate charter dates of 1746, 1754 and 1755.
(This seemingly minor difference of opinion assumes greater importance in the world of academia. Formal academic processions such as those at graduation ceremonies place visiting dignitaries and other officials in the order of their institution's founding dates, explaining why universities have sometimes used strained rationales to claim and defend dates as early as possible. The University of Pennsylvania changed its founding date in 1899, four years after elite universities in the United States agreed that academic processions would follow this age-based hierarchy. The revision in founding date was the result of a three-year campaign initiated by the University's "Alumni Register" magazine to make it older than Princeton for these processions.)
The argument used is that it is the common legal practice to date the founding of an institution from the date of founding for the oldest trust it administers. In this case, the oldest trust that the University of Pennsylvania administers was established in 1740. Historian Edward Potts Cheyney states that, "it might be considered a lawyer's date; it is a familiar legal practice in considering the date of any institution to seek out the oldest trust it administers." (The University still administers this trust in the funding of the Sadie Tanner Mossell Alexander University of Pennsylvania Partnership School.) He also points out that Harvard's founding date is merely the year in which the Massachusetts General Court resolved to establish a fund in a year's time for a "School or College". As well, Princeton claims its founding date as 1746--the date of its first charter. However, the exact words of the charter are unknown, the number and names of the trustees in the charter are unknown, and no known copy is extant. Though it is a common practice to use the dates of charter as the official date, the majority of the American Colonial Colleges do not have clear-cut dates of foundation. 
The history of the University of Pennsylvania as the successor organization to the College of Philadelphia and the University of the State of Pennsylvania is detailed by Penn's archives department.
In brief, in 1779 the College of Philadelphia was directed by provost William Smith. One might have expected it to evolve into the "University of the State of Pennsylvania" but this did not occur. "Since the Revolutionary state legislature felt that the board of trustees led by Provost Smith contained too many suspected loyalist sympathizers, they created a new board of trustees." Thus, the University of the State of Pennsylvania was created de novo. A schism occurred, with an attenuated College of Philadelphia continuing under Dr. Smith's direction. In 1791 Pennsylvania adopted a new state constitution which merged the College of Philadelphia and the University of the State of Pennsylvania into the "University of Pennsylvania," with a board of trustees made up of twelve men from each of the two parent institutions. "It is this institution and this board of trustees that has continued to this day."
On December 4, 1779, just seven days after the founding of the "University of the Commonwealth of Pennsylvania", an event occurred which William and Mary describes thus:
|“||Under the leadership of Thomas Jefferson, then governor of Virginia and a member of the Board of Visitors, William and Mary became a university. The grammar and divinity schools were discontinued, and a professorship of anatomy and medicine, and the first American chairs of law and police and modern languages were established. The elective system of studies was introduced at this time, the first such program in the United States.||”|
William and Mary has a published list of its first graduates (by Swem) available through its library.
(It is not clear from context, either above or in the paragraphs that follow, that the constitution meant to draw any semantic distinction between "college" and "university." )
The University of Pennsylvania uses "America's First University" as a slogan and has an official, succinct statement of the argument supporting this claim:
"Penn does not claim to be America's first college, but it is America's first University. In the Anglo-American model, a college, by definition, is a faculty whose subject specialization is in a single academic field. This is usually arts and sciences (often referred to as "liberal arts"), but may also be one of the professions: law, medicine, theology, etc. A university, by contrast, is the co-existence, under a single institutional umbrella, of more than one faculty. Penn founded the first medical school in America. In that year, therefore, Penn became "America's first university." If you wish to take the position that "first university" means first institution of higher learning with the name "university," Penn also qualifies as first. In 1779, the Pennsylvania state legislature conferred a new corporate charter upon the College of Philadelphia, renaming it the "University of the State of Pennsylvania" (in 1791 still another new charter granted Penn its current name). No other American institution of higher learning was named "University" before Penn. So whether you take the "de facto" position (1765) or the "de jure" position (1779), Penn is indeed "America's first university."