In mid-century New York, that is mid-19th-century New York, the idea of free public libraries was new and somewhat suspect. There were many comfortably-off, conservative-minded people who were unwilling to support an institution which might stir thoughts of betterment among the poor; or, worse, stir resentment at the impossible gap between themselves and the rich with whom they shared the city. Put another way: fearing the rise of organized labor and radical politics, these cautious power-wielding New Yorkers felt that a populace without books would be more malleable and easy to control than one that had learned to read, discuss, and thoughtfully consider matters.
This pessimistic set of mind is seen in the way the earliest free schools sought to inculcate unthinking obedience and patient endurance along with the three Rs. And it was is seen as well in the refusal of the city's alderman to give public funds so that libraries might be established for free lending of books to all comers. New York's earliest "public" libraries of the time were all privately funded and intended for the use restricted audiences. The best known among them served the needs of bibliophiles and scholars and they did not welcome as patrons men and women whose dress marked them as lower class.
All the same, not all of New York's power elite felt that public schools had to teach a specific Protestant morality or that "public" libraries were not supposed to serve the public. Some felt that New York deserved free circulating public libraries of its own simply because Boston (and other cities) had shown up New York by creating them. To these men the existence of a true public library was a matter of prestige, of civic pride. In this vein an editorial writer in the New York Times complained in this manner: "There is something humiliating in the reflection that our City and State have so long ignored this important subject, and that we are today so far in the rear of other States and countries."
At the same time some New Yorkers argued for the benefits that New York's citizens would receive from access to libraries. The arguments put forward by these progressives worked the conservatives' fears against them. As one of them put it, public libraries would help prevent the envy and grievances of poor working men and women from turning into class hatred. And they would help to prevent an ignorant populace from becoming easy prey for unscrupulous leaders. Libraries, this speaker said, would "thwart the efforts of the specious and designing men who undertake to use the motives and grievances of the people as aids to their own demagogic ambition and objects." He closed by an appeal to social order: "Let the rich men aid in this work of bridging over the chasm between themselves and the less fortunate or wealthy classes, and they would lay broader and deeper the foundation of society with a regulated liberty, in which the rich might not only enjoy the fruits of their toil and labors and intellects, but be safe from violence and crime and from the stupid hate and envy of those who have not succeeded as well as they in the battle of life." A bit later Carl Schurz expanded on the theme by stressing the importance of public libraries in a democracy. Libraries foster knowledge, he said, and it makes no sense to keep citizens ignorant when "our honor and greatness, the safety of our institutions, our whole social order, depend upon the intelligence and virtue with which the people govern themselves."
The progressives and those for whom it was a matter of civic pride seem to have made a significant advance when, in 1872, an editorial in the New York Times stated, quite baldly, "it would appear to be a proposition requiring no argument, that the City of New York greatly needs, and would largely profit by the early establishment of a Free Public Library." 
A few years later the first such library came into existence, but, surprisingly, it was "public" in the sense of "open to all" but not "public" in the sense of "publicly funded." The city's first free public lending library had its beginnings not in the construction of a large purpose-built structure for public use, but rather in the small-scale charitable effort of a few enlightened ladies belonging to the parish of the city's largest Episcopal church.
Here's the story: "The New York Free Circulating Library, now a part of the New York Public Library system, had its real beginning in 1879, in a sewing class in connection with the charitable work of Grace Church. The class was a small one, and as the girls showed an inclination to read cheap paper novels, one of the teachers proposed lending to each a book a week. In a short time other women became interested. About 500 books were collected, and a little library started in a room in Thirteenth Street, east of Fourth Avenue. It increased with wonderful strides, so that at the end of the first year about 1,200 books, all gifts, were on the shelves. The conclusion was reached that there was need for establishing a circulating library in various parts of the city, and this resulted in the formation, in 1880, of the New York Free Circulating Library. In March of that year the library was moved to two rooms at 36 Bond Street, where it remained until 1883, when it took possession of the entire building at 49 Bond Street (the present Bond Street Branch). From this time on branches were formed in various parts of the city, until [in 1901] the library was merged into the Public Library system."
This photo shows the building where the library had two rooms at 36 Bond St.
This shows the library at 49 Bond St. in 1885.
The Free Circulating Library was a great success. Usage was high and demand grew for branch libraries to be opened throughout the city.
This list names the nine branches that existed at the time the NYPL took them over in 1901. They appear in the order in which they were brought into the system.
- 1880, Bond St., 3rd location, 49 Bond St. May 1883
- 1884, Ottendorfer, 135 Second Ave.
- 1888, George Bruce, first location, 226 W. 42nd St.
- 1888, Jackson Sq, 251 W 13th St.
- 1892, Harlem, first location, 18 e. 125th St.
- 1893, Muhlenberg, first location, 220 6th Av.
- 1896, Bloomingdale, first location, 816 Amsterdam Ave.
- 1897, Riverside, first location, 261 W. 69th St.
- 1897, Yorkville, first location, 1523 2nd Ave.
This brief notice in the New York Times marks the opening of the first branch in 1880.
This photo shows school children at 49 Bond St. branch.
Here you can see catalog and some of the stacks at 49 Bond St.
This article in the New York Times is a brief status report from early in 1884.
IN AND ABOUT THE CITY; THE FREE LIBRARY PROJECT. FACTS DRAWN FROM THE EXPERIENCE OF THE CIRCULATING LIBRARY, New York Times, February 5, 1886
READING FOR THE PEOPLE - THE NEW-YORK FREE CIRCULATING LIBRARY. EX-PRESIDENT CLEVELAND AND OTHER CITIZENS ADDRESS A LARGE MEETING IN CHICKERING HALL, EX-PRESIDENT CLEVELAND AND OTHER CITIZENS ADDRESS A LARGE MEETING IN CHICKERING HALL, New York Times, March 7, 1890. First para: "Every seat in Chickering Hall was filled yesterday afternoon by an audience which assembled to listen to what several distinguished gentlemen had to say concerning the New-York Free Circulating Library, the institution which has conferred so many benefits upon the great number of persons who have claimed the use of its advantages."
LIBRARY WORK; Points in the Annual Report of the Free Circulating Library of New York, New York Times, March 25, 1899
BOOKS FOR THE PEOPLE - THE NEW-YORK FREE CIRCULATING LIBRARY. A GREAT DEAL OF WORK DONE WITH A VERY LITTLE MONEY -- FIGURES FOR THE PAST YEAR -- SOME NEEDS OF THE INSTITUTION; A GREAT DEAL OF WORK DONE WITH A VERY LITTLE MONEY -- FIGURES FOR THE PAST YEAR -- SOME NEEDS OF THE INSTITUTION. New York Times, January 15, 1892. First para: "The New-York Free Circulating Library has just issued its twelfth annual report. It sets forth the work done, and indicates what an extension of this work there might be if funds enough were forthcoming. The amount of good work done by this institution is well worth considering."
MANY BOOKS READ. - GROWTH OF THE NEW-YORK FREE CIRCULATING LIBRARY, New York Times, November 21, 1892. First para: "The annual report of the New-York Free Circulating Library for 1892 shows that the expenses of the institution during the fiscal year ending Sept. 30 were $27,708.33. All the departments of the library work show a large increase. The usefulness of the library has been greatly extended, and the whole condition improved."
List of libraries in 19th-century New York City in wikipedia
Statistics of Public, Society and School Libraries by the United States Office of Education (Govt. print. off. 1893).
Appleton's Dictionary of New York and its vicinity, with maps of New York and its environs (Appleton, 1898)
LIBRARIES FOR THE POOR; A MOVEMENT TO SUPPLY A MUCH NEEDED WANT, New York Times, January 21, 1882. First para: "The question of the need and the feasibility of establishing a number of well-organized and well-supplied free circulating libraries in this City was the matter which attracted a large and cultured audience last night to the ball of the Union League Club. The meeting was held under the auspices of the New-York Free Circulating Library, Bond-street."
A WELL-MANAGED INSTITUTION; FIRST ANNUAL REPORT OF THE NEW-YORK FREE CIRCULATING LIBRARY, New York Times, December 23, 1880. First para: "The first annual report of the New-York Free Circulating Library, which has just been issued, is invested with particular interest from the fact that the institution is the only free circulating library in this City. In this respect New-York was behind other cities, and this enterprise was undertaken without contemplating any burden on the tax-payer."
OUTGROWING ITS FACILITIES.; PROSPEROUS CONDITION OF THE NEW-YORK FREE CIRCULATING LIBRARY, New York Times, January 13, 1884. First para: "In the fourth annual report of the New York Free Circulating Library, for the year 1883, the Trustees say they have reason to believe that they will soon be called upon to assume charge of a free library which is being erected by a public-spirited citizen on the east side of the city."
REV. DR. JOHN HALL DEAD; The Pastor of the Fifth Avenue Presbyterian Church Expires Near Belfast, Ireland. HEART DISEASE THE CAUSE He Was on the Point of Returning to New York -- The Story of His Life of Unusual Activity. New York Times, September 18, 1898
The Free Circulating Library.; TWO NEW BRANCHES ESTABLISHED -- OTHER IMPORTANT CHANGES, New York Times, June 12, 1897
"Free Circulating Libraries," in American annual cyclopaedia and register of important events (D. Appleton and company, 1887)
"New York Free Public Libraries," in The Cosmopolitan, Vol. 3 (Schlicht & Field, 1887)
 You can see this attitude in debates, largely raised by Catholics, about the reading of the Protestant Bible to students in publicly-funded schools. As part of this argument the city superintendent of schools defended this reading as a means of procuring among students "order, quietude, neatness, punctuality, fidelity, industry, obedience, honor, truth, uprightness, deference to the wants, the rights, and conveniences of others." Here's more of his defense:
In all our Public Schools and departments, with very few and inconsiderable exceptions, the services of each day are commenced by the reading of selections from the Bible, by the Principal or some one of the School Officers, followed by the solemn and reverent repetition of the Lord's Prayer by all the pupils and teachers in concert, and by the singing of one or more appropriate hymns of Christian thanksgiving and praise. In many of them these devotional exercises are repeated at the close of the school. In all of them, without a solitary exception, at frequent and appropriate intervals during the day, songs, imbued with the purest principles of Christian morality, form a portion of the course of instruction, and are participated in by all the pupils. In all of them, without a solitary exception, lessons and precepts of virtue and Christian conduct are daily inculcated by the teachers, school officers, superintendents, or visitors; the fundamental principles of religion recognized and enforced; and the importance and necessity of strict honesty and integrity, undeviating truthfulness, frankness, sincerity, mutual affection and regard, obedience and respect to parents, and the conscientious and uniform observance of all the requisitions of a pure Christian morality, taught by precept and example. In all of them, the daily routine and discipline of the school are directly and powerfully adapted to the formation and perpetuation of habits of order, quietude, neatness, punctuality, fidelity, industry, obedience, honor, truth, uprightness, deference to the wants, the rights, and conveniences of others, and to the assiduous culture of the highest and noblest principles of action and conduct in all the varied relations of life. This is the character of the teachings of our Public Schools; these are the agencies and instrumentalities in daily operation within their walls; and no influences at variance with these are permitted, under any pretence, to find access or gain a footing among them. Neither the mind nor the heart of the child most religiously and scrupulously trained and disciplined in the domestic circle or the sanctuary of the Church, is exposed to the slightest contamination by the instructions or discipline of the school; while, on the other hand, every lesson of pure Christian morality or ethics, communicated in either of the former, is strengthened and confirmed by the pervading instruction and influence of the latter. -- Report of the City Superintendent of Schools to the Board of Education, New-York, Dec. 30, 1857; in Documents of the Board of Education of the City of New York (New York Board of Education, 1858) These were the Astor and Lenox libraries. See History of The New York Public Library on the NYPL web site and A Free Public Library and How to Get It, an editorial in the New York Times, January 28, 1872. Extract: "The Astor books were gathered for the single purpose of meeting the wants of the student, the critic, the scholar, and thus of feeding tastes already cultivated to the point of hunger. This purpose they are admirably fitted to serve. As a reference library, therefore, to be consulted on the premises, the Astor is not surpassed in this country." The writer says if the reluctant trustees would agree to expand the Astor to included a free, public, lending library, tens of thousands of New Yorkers "would flock to the newly-opened fountain of literature."
 The quote comes from Free Public Libraries, an editorial in the New York Times, January 14, 1872. Extract: "Other cities in our country have taken the lead of us a long way in this matter of a free distribution of books among the people. Notably, Boston has, in her splendid Public Library, met and answered successfully all the questions and obligations to the system as a practical scheme. [There are free libraries elsewhere in Massachusetts, in other states, and in foreign cities.] Even Italy has for several years enjoyed the advantages of popular circulating libraries, and at least thirty towns and cities have established libraries since 1861.... There is something humiliating in the reflection that our City and State have so long ignored this important subject, and that we are today so far in the rear of other States and countries." See also, for example, this from 1866: "Boston has a public library, where every resident of the city, without price or payment, can take home his couple of volumes from an immense and most valuable collection. New-York has a number of excellent libraries and reading rooms, (though not half enough for the wants of the City,) and yet their management and selection of late have not been satisfactory or sufficiently profitable. ... We find all sorts of barriers and formalities thrown up against readers, until many persons would rather do without a book or magazine, or purchase it, than laboriously extract it from the library. ... What we need, then, in New-York, is a greater hospitality and liberality in the existing libraries, as if they existed for the public and not the public for them." -- The Libraries of the City, editorial in the New York Times, June 10, 1866.
 The quotes come from a speech by Rev. Dr. John Hall, head of the Fifth Avenue Presbyterian Church, and one of New York's foremost religious leaders. The occasion was a meeting held at the Union League Club in support of the New-York Free Circulating Library, Bond Street. Here are some additional extracts of Hall's remarks from an account which appeared in the New York Times:
Idleness, vacant and unoccupied minds, added to temptations, were the conditions which produced and nurtured crime in the community, and to remedy this condition of things was a noble and worthy work. The present purpose was to give opportunity to those who do not now enjoy it of coming into contact with good books. -- not just "poetry, history, biography" but also fiction [even fiction, although many might disagree, the circulating library must provide it]... In the beginning this work he hoped the movers would bear in mind that it was not merely good things, but first-class things that were wanted for the people. In our land there was a large number of untutored, ignorant, stupid, and discontented men who were compelled to earn their living by severe labor, and who made known their grievances and discontent at times by appeals to violent feelings. Call them Communists, or the spirit which animated them Socialism, or whatever else they might, it was dangerous for the land in which it existed. Here was one way in which the higher and more favored classes might do much to thwart the efforts of the specious and designing men who undertake to use the motives and grievances of the people as aids to their own demagogic ambition and objects. Let the rich men aid in this work of bridging over the chasm between themselves and the less fortunate or wealthy classes, and they would lay broader and deeper the foundation of society with a regulated liberty, in which the rich might not only enjoy the fruits of their toil and labors and intellects, but be safe from violence and crime and from the stupid hate and envy of those who have not succeeded as well as they in the battle of life. -- LIBRARIES FOR THE POOR; A MOVEMENT TO SUPPLY A MUCH NEEDED WANT; Books to be Loaned Free of Cost. The question of the need and the feasibility of establishing a number of well-organized and well-supplied free circulating libraries in this City was the matter which attracted a large and cultured audience last night to the ball of the Union League Club. The meeting was held under the auspices of the New-York Free Circulating Library, Bond-street. New York Public Library: Ottendorfer Branch Dedication, in Speeches of Carl Schurz from Carl Schurz's papers in the Library of Congress; this is a printed proof with scattered hand-written edits by Schurz.
 The writer expressed a common view of "the poor" in adding that a lending library for the poor would make common books available to the lower classes and thereby "reach the hewers of wood and the drawers of water, the coal-heaver and the stevedore, the slop-shop seamstress and the household drudge, with the means of beguiling their listless leisure, feeding their hunger of mind where any exists, or tempting such mental appetite where it is latent."Free Reading for the People. New York Times, January 21, 1872
 This comes from a paperback pocketbook largely concerned with good works for the betterment of the poor: The better New York, A Practical Hand Book to the Resources and Progress of New York, by William Howe Tolman (The American Institute of Social Service, 1904). The sewing circle ladies were unusual in not requiring that their clients read morally uplifting literature. The other Grace Church good works seem to have had more missionary flavor. As for example the next entry in Tolman's book: "Strong and decided efforts are made day and night for the reformation of fallen women and young girls at the Florence Crittenden Mission, 21 Bleecker Street. Here the homeless may find a home and every incentive toward moral and educational advancement."
 The American Annual Cyclopaedia for ... gives usage statistics for the first five branches of the Free Circulating Library. The author reported that the five branches that existed in his day contained not much more than 100,000 volumes, and, "the eagerness with which these few books are sought for by the large number of readers who avail themselves of the privileges afforded, as shown by the statistics given below, is sufficient proof, if any were needed, of the use that would be made of a large free public library accessible to all, and of such a character as would place the city in the same rank in this respect as it maintains in other respects among the cities of the country." -- "Free Circulating Libraries" in American annual cyclopaedia and register of important events Vol. 26 (D. Appleton and company, 1887).
 The branch libraries were at first called distribution stations. Here's an account of the decision to create what we now call a library system: "It is the plan of the New York Free Circulating Library to establish 'stations' in different parts of the city for the distribution of books; stations each of which is, in fact, a complete library in itself, but all of which are under one general management — a sort of federation of libraries, as it were. This method, though having grave drawbacks, meets well the needs of a place the size of New York. To be obliged to go ten miles to change one's library book is a necessity always likely to weaken one's attachment to literature. On Bond street was established the first station, and now, though a second station of equal importance is established on Second avenue, it is generally referred to as a branch of the Bond-street Library." -- "NEW YORK FREE PUBLIC LIBRARIES" by Viola Roseboro in The Cosmopolitan Volume 3 (Schlicht & Field, 1887).