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When analysing a cup of coffee, a scientist might break it into solid (the cup), liquid (the coffee) and gas (the aroma). According to the Abhidhamma, a cup of coffee consists of rūpa such as hardness, roughness, heaviness, heat, visibility and odour. The Abhidhamma focus is on how the cup of coffee is impacting the senses. The focus of Buddhism is spiritual development and spiritual development involves the mind, so the Abhidhamma approach uses the mind as the starting point. This is why in the example of the cup of coffee, the Abhidhamma focuses on how the object impacts the senses.
The word “element” tells us to focus on the properties rather than the literal meaning of earth, water, fire or air. The Four Great Elements are experiences, Ultimate Realities, not concepts. The ancient Greeks and others used the same four elements to describe the composition of the world, but generally adopted the literal or symbolic meaning rather than Buddhist approach that focuses on properties. Rūpa are not abstract categories; they are the realities that appear in daily life.
Earth-element is the experience of hardness or softness, roughness or smoothness, heaviness or lightness. Here are some exercises to experience Earth-element. Be aware of hardness in the teeth; bite them together and feel how hard they are. Be aware of softness by pressing your tongue against the inside of your lips to feel its softness. Rub your tongue over the edge of your teeth or brush your hand over the skin of your arm and feel roughness. Experience smoothness by wetting your lips and rubbing your tongue over them, from side to side. Place one hand on top of the other in your lap and feel the heaviness of the top hand, or feel the heaviness of the head by bending it forward. Look for lightness by wagging a finger up and down and feeling its lightness. If something floats in the ocean, it is because the Earth-element in the ocean supports it. If a balloon floats, it is because the Earth-element in the air supports it.
Water-element is flowing and cohesion. Flowing and cohesion cannot be experienced, but it can be understood by the mind. To understand flowing, imagine the blood flowing through the blood vessels, the air flowing into the lungs, or heat flowing throughout the body. To understand cohesion, consider how the body is held together by the skin, flesh, and sinews. The blood is held inside by the skin, like water in a balloon. Without cohesion the body would fall into separate pieces. The force of gravity which keeps the body stuck to the earth is also cohesion.
Fire-element is the experience of heat and coldness. It is usually very easy to look for heat throughout the body. Feel the coldness of the breath as it enters the nostrils and then understand it throughout the body. The function of Fire-element is to ripen, mature, age and burn up. The Fire-element also aids in digestion.
Air-element is the experience of supporting and pushing. To experience supporting, relax your back so your body bends forward. Then straighten it and keep it straight. The force that keeps the body straight is supporting. The supporting Air-element ensures that all things maintain their shape and do not collapse. To experience pushing, be aware through the sense of touch of pushing in the centre of the head as you breathe in and out or experience the pushing as the chest expands or as the abdomen moves when breathing. Wherever there is movement, there is pushing. Though the function of Air-element is causing motion, movement is not the alteration between two stages of the same rūpa, but rather as the disappearance of one rūpa and the immediate emergence in its place of another rūpa. For example, in a movie theatre, many static frames are projected onto the screen each second and a sense of movement is created.
All groups of rūpa include the Four Great Elements. In a stone and a snowflake, the Four Great Elements exist in equal proportion, but of varying intensity. The intensity and the Mental Factor of Attention determine which experience body-consciousness recognizes first. For example, when you put your hand in water, the softness felt is Earth-element, the coolness felt is Fire-element and the pressure felt is Air-element. If the water is cold, then the Fire-element may be experienced first. If there is attention being paid to the sensation of pressure, then Air-element may be experienced first.
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This should be understood in the level of what is felt or experienced. Generally what you experience is element that dominates.
Earth - this is solidity, attractive forces, has fixed shape and space. When you bring you attention to your teeth the have a certain feel to it. Also the surface you site on has a texture you feel.
Water - this is attractive forces, binding, cohesion, mixing, flowing. If you bring your attention to you flesh it is what bonds you body. It holds it together. Also your saliva flows in the cavity of the mouth. The wood of the chair is help together but this you cannot feel butt you can perceive as the chair does not fall apart.
Fire - the heat or temperature, energy. Say you are famished then you are weak hence lacking vitality or heat element. In the case of a chair which was out in the sun you feel the heat.
Air - repulsive, expansion. Your breathing gives the sense of the Air element. If the chair is cushioned then there is some feel of cushion getting compressed in areas and expanding around you when you sink in.
Earth - when you experience painful sensation when it starts it starts small and thousands of other small sensation cluster together to give your the pain. (Gross solidified sensations.)
Water - sometimes you have itching and sensation like insects scrolling in your body. The sensation is running around off flowing. In some cases there are very pleasant sensations which soak your body like water being pored on your head of water absorbed by a blotting paper.
Fire - heat and cold you feel. Vibration in you body.
Air - pulsation and expansion and contraction you feel.
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The Manual of Abhidhamma says,
With His supernormal knowledge the Buddha analysed this so-called paramàõu and declared that it consists of paramatthas—ultimate entities which cannot further be subdivided.
The paramatthas are pañhavi, àpo, tejo, and vàyo. One must not understand that these elements are earth, water, fire and air as some Greek thinkers believed in the past.
Pañhavi means the element of extension, the substratum of matter. Without it objects cannot occupy space. The qualities of hardness and softness which are purely relative are two conditions of this particular element. It may be stated that this element is present in earth, water, fire and air. For instance, the water above is supported by water below. It is this element of extension in conjunction with the element of motion that produces the upward pressure. Heat or cold is the tejo element, while fluidity is the àpo element.
âpo is the element of cohesion. Unlike pañhavi it is intangible. It is this element that makes scattered particles of matter cohere and gives rise to the idea of ‘body’. When solid bodies are melted this element becomes more prominent in the resulting fluid. This element is found even in minute particles when solid bodies are reduced to powder. The element of extension and cohesion are so closely interrelated that when cohesion ceases extension disappears.
Tejo is the element of heat. Cold is also a form of tejo. Both heat and cold are included in tejo because they possess the power of maturing bodies. Tejo, in other words, is the vitalizing energy. Preservation and decay are also due to this element. Unlike the other three essentials of matter, this element has the power to regenerate matter by itself.
Inseparably connected with heat is vàyo, the element of motion. Movements are caused by this element. Motion is regarded as the force or the generator of heat. “Motion and heat in the material realm correspond respectively to consciousness and Kamma in the mental.”
These four elements coexist and are inseparable, but one may preponderate over another as, for instance, pañhavi in earth, àpo in water, tejo in fire, and vàyo in air.
They are also called Mahàbhåtas or Great Essentials because they are invariably found in all material substances ranging from the infinitesimally small cell to the most massive object.
According to this, both hardness and softness are conditions of Pañhavi. The main property of Pañhavi is "extension" i.e. the fact that it occupies space.
I guess that âpo is similar, i.e. that if pañhavi means that something can't be compressed (i.e. made to occupy a smaller space than it currently does), âpo means that it can't be pulled apart.
I don't know how to understand vàyo in an inanimate chair; in the body there's vàyo in breathing.
This answer seems to slightly contradict the other two answers:
Although you tagged this question abhidhamma the elements are also described in the suttas, in:
I think it would be a good idea to read the suttas because they provide context.