Mass is a fundamental unit in physics, whereas weight plays a decidedly minor role. It appears like a shadow for the term mass, only to disappear once the harsh light of scientific enquiry has revealed that the two are not identical. As mentioned above, Hewitt (1999, 49) defines mass as the "quantity of matter in an object". How does one measure the quantity or amount of matter? This is an ancient art as old as civilization itself which depends on (a) what is being measured and (b) for what purpose. Let us look at some specific examples from GDC commerce and SDC studies of motion, using typical objects such as food, precious metals, balls, and pucks.
        Perhaps the most primitive measure is to count the number of objects. Americans buy donuts and eggs by the dozen as long as they are of uniform size and quality. If the donuts or eggs are truly identical, then their number will be proportional to their SDC mass. To specify one is to specify the other. We can say number is functionally equivalent to mass with the unit of measurement being a single donut or egg. Likewise, in physics problems dealing with balls of a uniform shape (sphere), size, and material composition (steel), motion can be accurately analyzed simply by using the number of balls to represent their mass. A popular conversation piece which uses five identical balls suspended in a line to demonstrate the conservation of momentum is an example of such a simplification. Such a simple system of measurement is of limited use, since each unit (donut, egg, or ball) can only be applied to nearly identical objects.
        In order to measure to a wider range of objects with a single standard unit we could relax the conditions with respect to shape and try to measure volume. This works well for liquids and powders, as long as composition remains uniform. In Japan we buy liters of milk, while Americans buy quarts or gallons. With strict conditions on uniformity, size is proportional to and a valid measure of mass. Such a scale also allows us to measure mass along a continuum between the integers, since volume can be divided up and recombined to approximate any number of liters, quarts, or gallons. The motion of plastic pucks of uniform composition on an air plane can be analyzed using their volumes to represent mass. If their heights are uniform the square of the radius, since it would be proportional to volume and to mass, would be just as valid. Blocks of precious metal could be measured by size, but more complex shapes make volume quite difficult to measure. In addition the blocks might be hollow or filled inside with cheap metals so that an additional measure is necessary to verify internal composition.
        This brings us to GDC weight as a measure of amount. Heaviness is a property of objects perceptually almost as salient as number and size. Although you cannot see it, you sure can feel it. Weight, furthermore, is much simpler to measure than size. One simple machine for measuring weight is a balance scale with two platforms on each side. You compare weights by putting a known weight on one side and the object whose weight is to be measured on the other. When both sides balance evenly, the weights are equal. A slightly more sophisticated machine, such as many doctors used to use in their offices, might incorporate a lever. The principle is the same. After taking account of distances from the balance point, you are comparing a known weight with an unknown weight. More common today than a balance scale is a compression scale such as the bathroom scales often found in people's homes. You step on the scale so that the force of gravity pushes you against the top of the scale, while the floor and Earth support the bottom. This compresses springs inside. The amount of compression has been calibrated over a range of weights. People do not need to worry about shape or size. As long as the object to be measured fits on top of the scale, they simply place it there and look at the readout. To be really accurate and avoid the distortion of buoyancy provided by the sea of air that surrounds us, measurements can be made in a vacuum. For centuries weight has been the best measure of amounts of relatively dense objects from precious metals to meat at the supermarket. The measure of heaviness is so special that it even has its own verb--to weigh. This has been and continues to be an ideal system for Earth-bound people, who if they admit it still function from day to day with a flat-Earth mentality.
        In the 17th century came calculus and Newton's Laws of Motion. Physics tackled the mystery of orbiting planets and discovered gravity. A mechanism to explain gravity still aludes us. One popular theory suggest it's the work of gravitons. Some people jokingly say it's because the Earth sucks. Others don't care, they just think it is an attractive name for a condominium (Gravity Motoyama). Whatever gravity is, the planets' orbits seemed to be related to their heaviness. Since the planets could not be weighed as is done with common objects on Earth, this new kind of heaviness had to be measured by centrifugal forces or inertia, leading to the distinction of weight and mass. SDC weight is a measure of how heavy an object feels when the gravitational field of a second, usually a massive, object, like a planet, tries to accelerate it. SDC mass is a measure of how heavy an object feels when you try to accelerate it in the absence of, or horizontally to, the gravitational field. These measures of heaviness are exactly proportional when the gravitational field is held constant. The gravitational field, however, varies according to two factors: (a) the mass of the second object and (b) the distance away as measured from the center. Weight turned out to be more complicated in space than in the nearly uniform gravitational field on the surface of the Earth.
        We have reviewed four measures of amount: (a) number, (b) size, (c) GDC weight, and (d) SDC mass. Each has proven useful and continues to be used for certain scientific and commercial applications. If the physical world were composed of steel balls or a single atomic particle rather than neutron, protons, and electrons, mass could be assigned quantum numbers at a certain level of analysis. If all objects had uniform density, size would be an appropriate measure of amount. If the gravitational field were constant throughout the universe, weight and mass would be completely equivalent.

        Forces accelerate objects in specific directions. This makes forces, including forces of gravity and thus weight, vector quantities. Direction depends on the location of the second object, while magnitude depends on both masses and the distance of separation. Take the Earth as an example. How much does it weigh? If it were floating all by itself in space we could say its weight was zero. No gravity, no weight. To have gravity and weight we need a second object. To maximize the force of gravity it should have great mass or be very close (like the Moon). The Sun, which contains more than 99.8% of the mass of our solar system, is _{3.33 10}5 times as massive as Earth, where we usually weigh things, but is _{23.46 10}3 times as far from the center of the Earth as is the Earth's surface. So we can say the solar weight of the Earth is

597 kg 1022 kg x 3.33 105 __________________ (23.46 103) 2

= 361.53 1019 kg

This is the weight and the force that keeps the Earth in orbit around the Sun. The moon, with only 0.0123 as much mass as the Earth, is not nearly as massive as the Sun, but much closer to the Earth, only 60.27 times the Earth's radius away. So the lunar weight of the Earth is

597 kg 1022 kg x 0.0123 __________________ (60.27) 2

= 202.34 1017 kg

This is also the terrestrial weight of the Moon and the force that keeps the Moon in orbit around the Earth. This equality of weights follows from the fact that the force of gravity is a mutual attraction. Both bodies attract each other with the same force, which is proportional to the product of their masses. Their masses are different, but the product of their masses is the same (by the distributive property of multiplication). With this in mind, I might ask what is the weight of the Earth on the surface of this author. The answer is easy, the same as my weight on the surface of the Earth, about 80 kg.
        Like the Earth we each have a solar weight and lunar weight as well as our terrestrial weight. Why are we not aware of them? Is it all right to ignore gravitational contributions to our weight of all the planets except the one we are on? Yes, since the distance between a planet and the Sun are so great in comparison to the radius of each, solar gravitation on the surface of closest planet is only about 1% of the Mercury's own surface gravity and drops off sharply at increased distances (see Table 2).