Continental Grip
Origin of the term 'continental grip'
The earliest Google Books mention mention of "the Continental grip" appears in William Tilden, Racket Making by the House of Bancroft: With a Short Resume of the Game (1926):
There are three general styles of orthodox grips 1: The Eastern grips, used by such players as Vincent Richards, Brian I. C. Norton and myself, 2: The Western grip of W. M. Johnston and Howard Kinsey, and 3 : The English or Continental grip used by Rene Lacoste, and Col. A. R. F. Kingscote.
Since I am a believer in the Eastern grips, I will confine myself to them, with the passing comment that the Western grip is excellent for the forehand but unsound for the backhand. The English grip, with the low wrist on all ground strokes, has proved successful in the past, yet the broken line of the arm and hands does not commend itself to me, for broken line is weak under stress.
William "Big Bill" Tilden was the world's foremost tennis player of the 1920s; he grew up in Philadelphia, Pennsylvania, and lived there for much of his life. Vincent Richards was a prominent American tennis player during the 1920s and 1930s; he was a New Yorker, originally from the city of Yonkers. Brian Norton was a South African tennis players most prominent during the early 1920s.
W. M. "Little Bill" Johnston dominated U.S. tennis in the 1910s; he grew up in San Francisco, California. Howard Kinsey, whose most successful years in tennis were during the 1920s, was also from California.
René Lacoste, a major figure in international tennis during the 1920s, was born in Paris, France. Wikipedia identifies Algernon Kingsote as a British tennis player who flourished in the 1910s and 1920s; he was from Bangalore, India, and "learned playing tennis on the courts of the Château-d'Œx Club in Switzerland," according to Wikipedia.
In short, Tilden writing in 1926 associates the "Eastern grips" with players from Pennsylvania, New York, and South Africa; the "Western grip" with players from California; and "the English or Continental grip" with players from France and India (understood as part of the British Empire).
But this 1926 discussion is actually a revised version of Tilden's earlier book, The Art of Lawn Tennis, revised and enlarged edition (1922):
Racquet grip is a very essential part of stroke, because a faulty grip will ruin the finest serving. There is the so-called Western or Californian grip as typified by Maurice E. M'Loughlin, Willis E. Davis, and, to a slightly modified degree, Wm. M. Johnston, the American Champion. It is a natural grip for a top forehand drive. It is inherently weak for the backhand, as the only natural shot is a chop stroke.
The English grip, with the low wrist on all ground strokes, has proved very successful in the past. Yet the broken line of the arm and hand does not commend itself to me, as any broken line is weak under stress.
The Eastern American grip, which I advocate, is the English grip without the low wrist and broken line. To acquire the forehand grip, hold the racquet with the edge of the frame towards the ground and the face perpendicular, the handle towards the body, and "shake hands" with it, just as if you were greeting a friend. The handle settled comfortably and naturally into the hand, the line of the arm, hand, and racquet are one. The swing brings the racquet head on a line with the arm, and the whole racquet is merely an extension of it.
The backhand grip is a quarter circle turn of hand on the handle, bringing the hand on top of the handle and the knuckles directly up. The shot travels across the wrist.
Maurice McLoughlin, noted in this earlier book as a practitioner of the Western grip, was from Carson City, Nevada, and had his greatest success as a player in the early 1910s; he retired in 1919. Willis Davis, a star of the 1910s and early 1920s, was (like Bill Johnston) from San Francisco.
When and where did the term 'English grip' arise?
The term "English grip" is in fact considerably older than "Continental grip"—and was at the center of a battle of the tennis books in the early 1900s. First, from Reginald Doherty & Hugh Doherty, R.F. & H.L. Doherty on Lawn Tennis (1903):
The grip of the racket is a most important matter. It need not be necessarily be the same as ours, but we offer ours for what they are worth. Very nearly every first-class player has almost the same grip as ours.
The Dohertys do not give their preferred grip a name; it's simply the grip that "[v]ery nearly every first class player" uses. The Dohertys were excellent tennis players; they won the doubles championship in the English Covered Court Championships eight times in nine years from 1897 through 1905. In addition, R. F. Doherty won the men's singles championship at Wimbledon four straight years (1897–1900), and H. L. Doherty won it six years in a row between 1901 and 1906.
Controversy arose when an attorney named Pembroke Vaile wrote a book about tennis technique in which he criticized the standard grip used in England. From P. A. Vaile, Modern Lawn Tennis (1904):
I am giving illustrations of an English grip. These are the grips, fore and back hand, of H. S. Mahony, who is accounted, and probably rightly so, one of the best judges of lawn tennis in England.
I must say however that I cannot see any good points in the prevalent English grip which those shown by me do not possess.
Vaile does not give a categorical name for the grips he prefers, and he introduces the term "English grip" in the first instance with the indefinite article: "an English grip."
And finally, from F. W. Payn, Tennis Topics and Tactics (1907):
The grip shown in Plate 30 is almost the same as "the English grip" in "Great Lawn Tennis Players." In the [Lawn Tennis] Guide [a pamphlet that Vaile evidently wrote in 1906], however. Mr. Vaile found it necessary, in order to substantiate his attacks on "the English grip," to produce a grip which should be manifestly absurd for forehand driving, so he deserted his former photograph, which was not exaggerated enough, and produced a grip about the same as that of Plate 31 are.
...
In the section at the end of the Guide on "The Defective Hold of the Racquet," Mr. Vile first assumes that the Dohertys hold the racquet for the forehand drive with the thumb across and three-quarters of an inch beyond the handle of the bat on the other side, which I, for one, believe to be an exaggeration. Next he assumes that this is the common English grip for forehand strokes, which I know to be incorrect, and next he assumes that this is the reason why at present there are very few rising young players in England. There one has a specimen of the Vaile logic in a nutshell.
Elsewhere in the same book, Payn refers to "Mr. Vaile's oft-repeated assertions about the imaginary English grip." Frederick Payn was a good tennis player who reached his peak of success during the period 1900–1902, when he reached the quarterfinals at Wimbledon twice. He, the Dohertys, and P.A. Vailes were all Englishmen.
The great English tennis player Fred Perry, who was born (in 1909) in Stockport (Greater Manchester) and lived in Lancashire and Merseyside while growing up, was perhaps the dominant player of the 1930s—but he was far too late to the scene for his grip to be the original object of the term "English grip" or the later term "continental grip."
Conclusion
From the sources I consulted, it appears that (1) the expressions "Eastern grip[s]," "Western grip" originated in the United States, where the distinction between Eastern and Western grips makes geographical sense; (2) the expression "English grip" arose in connection with a critique (by a person of dubious qualifications) of the normal grip used by advanced English players in the first decade of the 1900s; (3) all three expressions are at least a bit older than the expression "Continental grip"; and (4) the sense of "Continental grip" (in use since at least 1926)—and its predecessor, "English grip" (in use since at least 1904)—is that, unlike the two main grips putatively of U.S. origin, it was originally popularized in Europe (viewed, in the broad U.S. way, as including the British Isles).
But given the fact that the original 1904–1907 squabble over "the English grip" was conducted entirely by Englishmen, it isn't clear to me whether "the English grip" (as Bill Tilden uses the term in 1922) and "the English or Continental grip" as Tilden uses it in 1926) refers to the Doherty-style grip of 1903, to the Vaile-recommended grip of 1904, or to some third grip developed between 1907 and 1922 (and perhaps "exported" from England to continental Europe at some point).