Shortly after the invention of the telegraph in the late 1830s, inventors attempted to develop a way to transmit sounds along electrical wires. But the process took decades to achieve full success. The earlier Reis telephone, mentioned in this review, used an intermittent electrical contact that rapidly connected and disconnected in order to create electrical impulses to transmit sounds, but it never achieved the fidelity needed to re-create understandable speech. Alexander Graham Bells' breakthrough was the use of "undulatory current" -- now known as "amplitude modulation" -- in order to send the sound currents.
 
Journal of the Society of Telegraph Engineers, 1876, pages 519-525:

BELL'S  ARTICULATING  TELEPHONE.

    Attempts have been made for many years past to transmit musical or articulate sounds to a distance by means of electrical communication, and some of the early experiments of the late Sir Charles Wheatstone were accompanied with so much success that it was hoped that a time would come when an instrument might be constructed not only to register graphically certain audible sounds but to produce upon a diagram a set of signs by which the sounds of the human voice could be recorded; in other words, that it might become possible to construct an automatic reporter; and in the Loan Collection of scientific apparatus at South Kensington may be seen several instruments bearing upon these researches, and in which the vowel sounds are recorded by a series of distinctive curves.
    In the year 1860, Philipp Reiss, of Friedrichsdorf, near Homburg, following the researches of Wertheim, Marian, and Henry upon the production of sounds by electricity, invented the telephone which bears his name, and which also may be seen at South Kensington. The telephone of Reiss is of two parts; a transmitting instrument and a receiver. The former consists essentially of a stretched membrane, which, by vibrating in unison with the impulses it receives from musical sounds played near it, transforms those impulses into a series of electrical currents by a simple make-and-break arrangement, and these currents acting on the receiving instrument, which may be hundreds of miles distant, reproduce the corresponding notes, so that a tune played at one station can be distinctly heard at the other.
    The receiving instrument is founded upon the well-known phenomenon discovered by Page in the year 1837, that a distinct sound accompanies the demagnetisation of an iron bar placed in an electro-magnetic helix. It consists of a soft iron bar about the size of a knitting needle, surrounded by a helix of wire which forms part of a voltaic circuit with the transmitting instrument, and for intensifying the effect both instruments are provided with sounding-boards, or resonators. From the above description it will be seen that if a note which makes (say) one hundred vibrations per second be sounded in the neighbourhood of the transmitting instrument, its membrane will make one hundred corresponding vibrations, making and breaking the voltaic current one hundred times, and producing one hundred demagnetisations in the receiving instrument for every second of time, so that exactly the same note that was sounded in the transmitter will be audible at the distant station. It is obvious that the duration of, and time between, two notes must be identical at both ends of the conducting wire, and thus is reproduced automatically and without a possibility of error the elements which make up melody, viz., correctness of note combined with measure of time. Fig. 1
    Following Reiss in Germany, Elisha Gray in America constructed in 1874 his far more perfect electric telephone, in which the transmitting instrument consists of a vibrating reed, which is at once a note-producer and a rheotome or contact-breaker. It is tuned like the reed of a harmonium to its proper note, and when adjusted can only transmit to the receiving instrument the number of currents per second corresponding to the vibrations producing its note. Elisha Gray's receiving instrument is electrically similar in principle to that of Reiss, but consists of a horse-shoe electro-magnet, mounted upon a wooden sounding-box or resonator, with a heavy armature attached to its poles. The transmitting instrument is provided with a key-board similar to that of a harmonium, and each note has its corresponding key and vibrating reed.
    The same inventor has since introduced his splendidly worked out telephonic telegraph, by which four or more distinct messages may be transmitted in the Morse code simultaneously along a single wire. This apparatus depends for its principle upon having a vibrator at the receiving station, tuned so as to be affected only by its corresponding transmitter at the sending station, and thus the receiving instruments along a line of wire have the power of selecting those messages intended for themselves and letting all others pass. This has also been accomplished by a Danish engineer, M. Paul Lacour, who employs vibratory tuning-forks for transmitting the impulses, and a series of corresponding tuning-forks, each arm of which is inclosed in a magnetic helix for the selecting instrument. This selecting instrument can be used either as a receiving telephone, or by being employed as an intermediate relay may transmit the signals to ordinary telegraph instruments.
    We give herewith illustrations of the transmitting and receiving instruments of Mr. Graham Bell's articulating telephone, by which the sound of the human voice may be transmitted by electricity along a telegraph line, and heard, as a voice, at the other end.
    The articulating telephone of Mr. Graham Bell, like those of Reiss and Gray, consists of two parts, a transmitting instrument and a receiver, and one cannot but be struck at the extreme simplicity of both instruments, so simple indeed that were it not for the high authority of Sir William Thomson one might be pardoned at entertaining some doubts of their capability of producing such marvellous results. Fig. 2
    The transmitting instrument, which is represented in fig. 1, consists of a horizontal electro-magnet, attached to a pillar about 2 inches above a horizontal mahogany stand; in front of the poles of this magnet--or, more correctly speaking, magneto electric inductor--is fixed to the stand in a vertical plane a circular brass ring, over which is stretched a membrane, carrying at its centre a small oblong piece of soft iron, which plays in front of the inductor magnet whenever the membrane is in a state of vibration. This membrane can be tightened like a drum by the three mill-headed screws shown in the drawing. The ends of the coil surrounding the magnet terminate in two binding-screws, by which the instrument is put in circuit with the receiving instrument, which is shown in fig. 2. This instrument is nothing more than one of the tubular electro-magnets invented by M. Niclès in the year 1862, but which has been re-invented under various fancy names several times since. It consists of a vertical bar electro-magnet inclosed in a tube of soft iron, by which its magnetic field is condensed and its attractive power within that area increased. Over this is fixed, attached by a screw at a point near its circumference, a thin sheet iron armature of the thickness of a sheet of cartridge paper, and this when under the influence of the transmitted currents acts partly as a vibrator and partly as a resonator. The magnet with its armature is mounted upon a little bridge which is attached to a mahogany stand similar to that of the transmitting instrument.
    The action of the apparatus is as follows: When a note or a word is sounded into the mouthpiece of the transmitter, its membrane vibrates in unison with the sound, and in doing so carries the soft iron inductor attached to it backwards and forwards in presence of the electro-magnet, inducing a series of magneto-electric currents in its surrounding helix, which are transmitted by the conducting wire to the receiving instrument, and a corresponding vibration is therefore set up in the thin iron armature sufficient to produce sonorous vibrations by which articulated words can be distinctly and clearly recognised.
    In all previous attempts at producing this result the vibrations were produced by a make-and-break arrangement, so that while the number of vibrations per second as well as the time measures were correctly transmitted there was no variation in the strength of the current, whereby the quality of tone was also recorded. This defect did not prevent the transmission of pure musical notes, nor even the discord produced by a mixture of them, but the complicated variations of tone, of quality, and of modulation, which make up the human voice, required something more than a mere isochronism of vibratory impulses.
    In Mr. Bell's apparatus not only are the vibrations in the receiving instrument isochronous with those of the transmitting membrane, but they are at the same time similar in quality to the sound producing them, for, the currents being induced by an inductor vibrating with the voice, differences of amplitude of vibrations cause differences in strength of the impulses, and the articulate sound as of a person speaking is produced at the other end.
    Of the capabilities of this very beautiful invention, we cannot give them better than in the words of an ear witness, and no less an authority than Sir William Thomson, who in his opening address to Section A at the British Association at Glasgow thus referred to it:
    "In the Canadian Department I heard 'To be or not to be . . . there's the rub,' through an electric telegraph wire; but, scorning monosyllables, the electric articulation rose to higher flights, and gave me passages taken at random from the New York newspapers: 'S. S. Cox has arrived' (I failed to make out the 'S. S. Cox'); 'the City of New York;' 'Senator Morton;' 'the Senate has resolved to print a thousand extra copies;' 'the Americans in London have resolved to celebrate the coming 4th of July.' All this my own ears heard, spoken to me with unmistakable distinctness by the then circular disc armature of just such another little electro-magnet as this which I hold in my hand. The words were shouted with a clear and loud voice by my colleague judge, Professor Watson, at the far end of the telegraph wire, holding his mouth close to a stretched membrane, such as you see before you here, carrying a little piece of soft iron, which was thus made to perform in the neighbourhood of an electro-magnet, in circuit with the line, motions proportional to the sonorific motions of the air. This, the greatest by far of all the marvels of the electric telegraph, is due to a young countryman of our own, Mr. Graham Bell, of Edinburgh and Montreal and Boston, now becoming a naturalised citizen of the United States. Who can but admire the hardihood of invention which devised such very slight means to realise the mathematical conception, that, if electricity is to convey all the delicacies of quality which distinguish articulate speech, the strength of its current must vary continuously and as nearly as may be in simple proportion to the velocity of a particle of air engaged in constituting the sound."
--Engineering.