Following the Path of Discovery
Repeat Famous Experiments and Inventions
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Simple lenses made from rock crystal and their properties were known well before the invention of the modern optical telescope.
Ptolemy (in his work Optics written in the 2nd century AD) wrote about the properties of light including reflection, refraction, and color.
During the 10th century, Ibn Sahl, a Muslim mathematician of the Islamic Golden Age, in his treatise On Burning Mirrors and Lenses sets out his understanding of how curved mirrors and lenses bend and focus light. Ibn Sahl is credited with first discovering the law of refraction, usually called Snell's law. He used the law of refraction to derive lens shapes that focus light with no geometric aberrations, known as anaclastic lenses. 
It was approximately from the 12th century in Europe that 'reading stones' (magnifying lenses placed on the reading material) were well documented—as well as the use of lenses as burning glasses. It is generally considered that spectacles for correcting long sightedness with convex lenses were invented in Northern Italy in the late 13th to early 14th century, and the invention of the use of concave lenses to correct near-sightedness is ascribed to Nicholas of Cusa in 1451. Thus, early knowledge of lenses and the availability of lenses for spectacles from the 13th century onwards through the 16th century means that it was possible for many individuals to discover the principles of a telescope using a combination of concave or concave and convex lenses; in the 13th century, Robert Grosseteste wrote several scientific treatises between 1230 and 1235, including De Iride (Concerning the Rainbow), in which he said:
"This part of optics, when well understood, shows us how we may make things a very long distance off appear as if placed very close, and large near things appear very small, and how we may make small things placed at a distance appear any size we want, so that it may be possible for us to read the smallest letters at incredible distances..."  
Pre 17th Century Developments
There is some documentary evidence, but no surviving designs or physical evidence, that the principles of telescopes were known in the late 16th century. Writings by John Dee and Thomas Digges in England in 1570 and 1571, respectively ascribe the use of both reflecting and refracting telescopes to Thomas' father Leonard Digges. They may have been experimental devices and were never widely reported or reproduced. Thomas Digges describes his father's device (In the preface to his 1591 Pantometria) as follows:
"But to leave these celestial causes and things done of antiquity long ago, my father by his continual painful [painstaking] practices, assisted with demonstrations Mathematical, was able, and sundry times hath by proportional Glasses duly situate in convenient angles, not only discovered things far off, read letters, numbered pieces of money with the very coin and superscription thereof, cast by some of his friends of purpose upon downs in open fields, but also seven miles off declared what hath been done at that instant in private places."
Although Digges may have created a rudimentary instrument involving lenses and mirrors, the optical performance required to see the details of coins lying about in fields, or private activities seven miles away, was far beyond the technology of the time.
In Italy, Giambattista della Porta also described a possible telescope as early as 1586 when he wrote in a letter, "...to make glasses that can recognize a man several miles away." In his Natural Magic published in 1589 he wrote:
"With a Concave lens you shall see small things afar off very clearly. With a Convex lens, things nearer to be greater, but more obscurely. If you know how to fit them both together, you shall see both things afar off, and things near hand, both greater and clearly." 
Della Porta was preoccupied with other things at the time and thought the idea of a "telescope" unimportant.
The Modern Telescope
The practical exploitation of the instrument was certainly achieved and came to public attention in the Netherlands at about 1608, but the credit of the original invention has been claimed on behalf of three individuals: Hans Lippershey and Sacharias Jansen—spectacle-makers in Middelburg, and Jacob Metius of Alkmaar (also known as Jacob Adriaanszoon).
Hans Lippershey was credited with creating and disseminating designs for the first practical telescope. On October 2, 1608 he applied for a patent for an instrument "for seeing things far away as if they were nearby," (beating Jacob Metius's patent by a few weeks). Lippershey failed to receive a patent since the same claim for invention had been made by other spectacle-makers. Lippershey was handsomely rewarded by the Dutch government for copies of his design. Sacharias Jansen's design for a telescope may have pre-dated Lippershey and Metius, but the invention was never widely publicized. Hans Lippershey is the earliest person documented to have applied for a patent for the device.
The original Dutch telescopes were composed of a convex and a concave lens - telescopes that are constructed this way do not invert the image. Lippershey's original design had only 3x magnification. Telescopes seem to have been made in the Netherlands in considerable numbers soon after the date of their invention, and rapidly found their way all over Europe.
Galileo Galilei’s Contribution
Galileo not only made important improvements to the telescope but was the first to use the new invented device for astronomical observations and by that he sparked the scientific revolution of the 17th century.
Galileo devoted his time to improving and perfecting the telescope and soon succeeded in producing telescopes of greatly increased power. His first telescope magnified three times (like Lippershey’s), but he soon made instruments which magnified eight times and finally, one that magnified about thirty times. With this last instrument, only about 120 centimeters long and had an objective diameter of 5 centimeters, he discovered in 1610 the satellites of Jupiter and soon afterwards the spots on the sun, the phases of Venus, and the hills and valleys on the Moon. In this last achievement he now appears to have been preceded by Thomas Harriot who made the first drawings of the moon with the aid of a telescope in July 1609.
Galileo further demonstrated the revolution of the satellites of Jupiter around the planet and gave rough predictions of their configurations, proved the rotation of the Sun on its axis, established the general truth of the Copernican system as compared with that of Ptolemy, and fairly routed the fanciful dogmas of the philosophers.
Galileo's instrument was the first to be given the name "telescope". The name was invented by the Greek poet/theologian Giovanni Demisiani at a banquet held on April 14, 1611 by Prince Federico Cesi to make Galileo Galilei a member of the Accademia dei Lincei. The word was created from the Greek tele = 'far' and skopein = 'to look or see'; teleskopos = 'far-seeing'.
These brilliant achievements, together with Galileo's immense improvement of the instrument, overshadowed to a great degree the credit due to the original inventor, and led to the universal adoption of the name of the Galilean telescope for the form of the instrument invented by Lippershey.
Build your own Galilean Telescope
The refracting telescope, sometimes referred to as a refractor telescope, is made up of concave and convex lenses that allow the light to be refracted by the lenses and as a consequence the images appear brighter and larger.
There are different types of refracting telescopes. The Galilean Telescope - named after its creator and was one of the first versions; the Keplerian Telescope, invented by Johannes Kepler in 1611 as an improvement on Galileo's design and others.
In this section we are going to build the Galilean type.
The advantages of Refracting Telescopes are easiness of use; simplicity of design and maintenance; good for lunar, planetary and distant terrestrial viewing.
The disadvantages are small apertures (the diameter of a telescope's main lens), typically 3 to 5 inches; chromatic aberration (false colour) and therefore is less suited for viewing small and faint deep sky objects such as distant galaxies and nebulae.
A simple Galilean telescope consists of a convex lens for the objective and a concave lens for the eyepiece. The simplest method to make the tube is to use two cardboard tubes, one that slides inside the other.
In order to determine how long the tube needs to be, take in account that the focal point of the ocular lens (eyepiece) is the same as the focal point for the objective lens. This idea is demonstrated in the following image.
For more information and building instruction consult the following links (you can always google for "building / making galilean telescope" ).
Now, after you are done, you can follow in the steps of Galileo and direct your new telescope to the sky.
2. B. Wolf, "Geometry and dynamics in refracting systems", European Journal of Physics 16, p. 14-20, 1995.
3. R. Rashed, "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses", Isis 81, p. 464–491, 1990.
5. Kriss, Timothy C.; Kriss, Vesna Martich (April 1998). "History of the Operating Microscope: From Magnifying Glass to Microneurosurgery". Neurosurgery 42 (4): 899–907.
7. Giambattista della Porta, (2005), Natural Magick, page 339. NuVision Publications, LLC.
9. Phyllis Allen, Problems Connected with the Development of the Telescope (1609-1687), Isis, Vol. 34, No. 4 (Spring, 1943), pp. 302-311 (article consists of 10 pages)
13. The mechanics of the telescope, Burrell, E. P. Popular Astronomy, Vol. 39, p.310