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The Invention of the Compound Microscope
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 mean that it was possible for many individuals to discover the principles of a microscope or a telescope using concave and convex lenses.  
The first microscope to be developed was the optical microscope, although the original inventor is not easy to identify. An early microscope was made in 1590 in Middelburg, Netherlands when two Dutch lens grinders Hans and Zacharias Janssen (father and son) made a microscope by placing two lenses in a tube.  . Zacharias Janssen is also believed to invent an early version of a telescope. 
Hans Lippershey (who developed an early telescope) is also believed to build an early version of a compound microscope (using more than one lens).  It’s only normal that the guys which toyed with early microscopes tried also to invent telescopes and vice versa.
Another favorite for the title of 'inventor of the microscope' was Galileo Galilei. He developed a compound microscope (Galileo had called it the "occhiolino" or "little eye") with a convex and a concave lens in 1609 (about the same time he also build his telescopes). Galileo's microscope was celebrated in the Accademia dei Lincei in 1624 and was the first such device to be given the name "microscope" a year later by Giovanni Faber. Faber coined the name from the Greek words micron meaning "small", and skopein meaning "to look at", a name meant to be analogous with "telescope". 
In 1619 Cornelius Drebbel designed and built telescopes and microscopes and was involved in a building project for the Duke of Buckingham. William Boreel, the Dutch Ambassador to England, mentions the microscope that was developed by Drebbel. Drebbel became famous for his invention in 1621 of a microscope with two convex lenses. Several authors, including Christiaan Huygens assign the invention of the compound microscope to Drebbel. 
Christiaan Huygens, another Dutchman, developed a simple 2-lens ocular system in the late 17th century that was achromatically corrected (use of lenses that correct distortion of color and shape), and therefore a huge step forward in microscope development. The Huygens ocular is still being produced to this day, but suffers from a small field size, and other minor problems. 
A precursor of modern microscopy and microbiology was Robert Hooke’s book Micrographia, detailing then thirty year-old Hooke's observations through various lenses. Published in September 1665, the first major publication of the Royal Society, it was the first scientific best-seller, inspiring a wide public interest in the new science of microscopy. Hooke made the first recorded microscopic observation ever - the fruiting bodies of molds, in 1665. Hooke was also the first researcher to use a microscope to observe the structure of plants. He found them to consist of tiny walled "chambers" that he called 'cells'. Hooke also made a copy of Leeuwenhoek's light simple microscope and then improved upon his design. 
Until 1800, compound microscopes designed by Hooke and others were limited to magnifications of 30x to 50x, and their images exhibited blurry edges (spherical aberration) and rainbowlike distortions (chromatic aberration). The compound microscope of the 17th century was inefficient due to difficulties in configuring multiple lenses and the time was ripe for a new device, much more simple and also much more powerful - Leeuwenhoek's simple microscopes that were using only one small bi-convex lens (see below).
Anton van Leeuwenhoek Contributions
Hooke made the first recorded microscopic observation but Van Leeuwenhoek was the first to observe single-celled organisms like microbes. Van Leeuwenhoek is also credited with the invention of the simple microscope which uses only one magnifying lens, which was much better that the compound microscope at the time.
In 1676, Antonie van Leeuwenhoek observed bacteria and other microorganisms in water, the first bacteria observed by man, using a single-lens microscope of his own design.  
Anton van Leeuwenhoek (1632–1723) is credited with bringing the microscope to the attention of biologists, even though simple magnifying lenses were already being produced in the 16th century. Van Leeuwenhoek's home-made microscopes were very small simple instruments, with a single, yet strong lens (up to 500X in comparison to the 50x of contemporary compound microscopes).  They were awkward in use, but enabled van Leeuwenhoek to see detailed images. It took about 150 years of optical development before the compound microscope was able to provide the same quality image as van Leeuwenhoek's simple microscopes, due to difficulties in configuring multiple lenses. Still, despite widespread claims, van Leeuwenhoek is not the inventor of the microscope but the inventor of the simple microscope which uses one magnifying lens.
However, Leeuwenhoek is commonly known as "the Father of Microscopy and Microbiology", and considered to be the first microbiologist. He is best known for his work on the improvement of the microscope and for his contributions towards the establishment of microbiology. Using his handcrafted microscopes he was the first to observe and describe single celled organisms, which he originally referred to as animalcules, and which we now refer to as microorganisms. He was also the first to record microscopic observations of muscle fibers, bacteria, spermatozoa and blood flow in capillaries (small blood vessels). Van Leeuwenhoek did not author any books, although he did write many letters.
Van Leeuwenhoek's interest in microscopes and a familiarity with glass processing led to one of the most significant developments in the history of science. By placing the middle of a small rod of soda lime glass in a hot flame, Van Leeuwenhoek could pull the hot section apart like taffy to create two long whiskers of glass. By then reinserting the end of one whisker into the flame, he could create a very small, high-quality glass sphere. These spheres became the lenses of his microscopes, with the smallest spheres providing the highest magnifications. An experienced businessman, Leeuwenhoek realized that if his simple method for creating the critically important lens was revealed, the scientific community of his time would likely disregard or even forget his role in microscopy. He therefore allowed others to believe that he was laboriously spending most of his nights and free time grinding increasingly tiny lenses to use in his microscopes. Leeuwenhoek constructed hundreds of microscopes and nourished a passion for building new microscope whenever he chanced upon an interesting specimen that he wanted to preserve. 
Van Leeuwenhoek ground more than 500 optical lenses. He also created at least 250 microscopes, of differing types, of which only nine survived. His microscopes were made of silver or copper frames, holding hand-ground lenses. Those that have survived are capable of magnification up to 275 times. It is suspected that Van Leeuwenhoek possessed some microscopes which could magnify up to 500 times. Although he has been widely regarded as a dilettante or amateur, his scientific research was of remarkably high quality. 
Van Leeuwenhoek's main discoveries are:
Topics of Interest
The optical microscope, often referred to as the "light microscope", is a type of microscope which uses visible light and a system of lenses to magnify images of small samples.
A simple microscope is a microscope that uses only one lens for magnification, and is the original design of the light microscope like Van Leeuwenhoek's microscopes which consisted of a small, single converging lens mounted on a brass plate, with a screw mechanism to hold the sample or specimen to be examined. Though now considered primitive, the use of a single, convex lens for viewing is still found in simple magnification devices, such as the magnifying glass, and the loupe.
A compound microscope is a microscope which uses multiple lenses to collect light from the sample and then a separate set of lenses to focus the light into the eye or camera. Compound microscopes are heavier, larger and more expensive than simple microscopes due to the increased number of lenses used in construction. The main advantages of multiple lenses are improved resolution and sample contrast, reduced chromatic aberration and exchangeable objective lenses to adjust the magnification. A compound microscope also makes more advanced illumination setups, such as phase contrast.
An electron microscope is a type of microscope that produces an electronically-magnified image of a specimen for detailed observation. The electron microscope (EM) uses a particle beam of electrons to illuminate the specimen and create a magnified image of it. The microscope has a greater resolving power than a light-powered optical microscope, because it uses electrons that have wavelengths about 100,000 times shorter than visible light (photons), and can achieve magnifications of up to 2,000,000x, whereas light microscopes are limited to 2000x magnification.
Build Your Own Leeuwenhoek Microscope Replica
In order to understand better how a Leeuwenhoek microscope works, try to build a replica for yourself which is not complicated.
Leeuwenhoek has probably made over 500 microscopes of which a few survived. Basically, Leeuwenhoek's instruments were simply powerful magnifying glasses, not compound microscopes of the type used today. A Leeuwenhoek microscope is a very simple device, using only one convex lens (1 to 2mm in diameter),  mounted in a tiny hole in the brass plate that makes up the body of the instrument. The specimen was mounted on a sharp point that sticks up in front of the lens, and its position and focus could be adjusted by turning two screws. The entire instrument was only 3-4 inches long, and had to be held up close to the eye; it required good lighting and great patience to use. But nevertheless, it was enough for its creator to be firmly established in history as one of the first and most important explorers of the microscopic world.
Consult the following links for building instructions and further information.
When you are done, you can follow in the steps of Anton van Leeuwenhoek and direct your new microscope to a drop of water.